{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-06T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":46054,"title":"Count trailing zeros in a primorial","description":"\u003chttps://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial Cody Problem 44068\u003e asked us to count the  trailing zeros in a factorial. This problem deals with the  _\u003chttps://mathworld.wolfram.com/Primorial.html primorial\u003e_. If p_n is the nth prime number, then the primorial p_n# is the product of the prime numbers up to and including p_n. For example, if n = 5, then p_5# = 2*3*5*7*11 = 2310, which has one trailing zero. \r\n\r\nCount the trailing zeros in the primorial p_n#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.* \r\n\r\nPlease make your code general because I might add more tests later. \r\n\r\n*;-)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 205px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 102.5px; transform-origin: 407px 102.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.75px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.8833px; text-align: left; transform-origin: 384px 31.8833px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 44068\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 238.833px 7.91667px; transform-origin: 238.833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asked us to count the trailing zeros in a factorial. This problem deals with the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/Primorial.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eprimorial\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 7.91667px; transform-origin: 9.71667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6167px 7.91667px; transform-origin: 20.6167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.28333px 7.91667px; transform-origin: 7.28333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime number, then the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 178.55px 7.91667px; transform-origin: 178.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# is the product of the prime numbers up to and including \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 52.1167px 7.91667px; transform-origin: 52.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example, if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.85px 7.91667px; transform-origin: 29.85px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 5, then \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_5\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.8667px 7.91667px; transform-origin: 11.8667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# = \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2*3*5*7*11\" style=\"width: 92px; height: 18px;\" width=\"92\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 112.233px 7.91667px; transform-origin: 112.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 2310, which has one trailing zero.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.1333px; text-align: left; transform-origin: 384px 21.1333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 121.767px 7.91667px; transform-origin: 121.767px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCount the trailing zeros in the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 254.467px 7.91667px; transform-origin: 254.467px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.*\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 215.1px 7.91667px; transform-origin: 215.1px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ePlease make your code general because I might add more tests later.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1833px 7.91667px; transform-origin: 29.1833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problems 46057 \"Find the last non-zero digit in a primorial\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.91667px; transform-origin: 15.5667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46060 \"Identify primorial primes\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.33333px 7.91667px; transform-origin: 9.33333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e*;-)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = primorialTrailingZeros(n)\r\n  y = f(n);\r\nend","test_suite":"%%\r\nn = 1;\r\ny_correct = 0;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 2;\r\ny_correct = 0;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 5;\r\ny_correct = 1;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 10;\r\ny_correct = sin((n+3)*pi/2);\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 100;\r\ny_correct = sum(1./2.^(1:n));\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 1e4;\r\ny_correct = besselj(mod(n,10),prod(num2str(n)'-'0'));\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 1e6;\r\nf = [factor(log10(n)) 3 7 7 89];\r\ny_correct = length(primes(n))/prod(f);\r\nassert(isequal(primorialTrailingZeros(n),y_correct))","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":83,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-26T15:54:06.000Z","updated_at":"2025-12-09T22:52:30.000Z","published_at":"2020-07-26T16:20:27.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 44068\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asked us to count the trailing zeros in a factorial. This problem deals with the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/Primorial.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eprimorial\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime number, then the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# is the product of the prime numbers up to and including \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example, if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 5, then \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# = \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2*3*5*7*11\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2\\\\cdot3\\\\cdot5\\\\cdot7\\\\cdot11\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 2310, which has one trailing zero.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCount the trailing zeros in the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease make your code general because I might add more tests later.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems 46057 \\\"Find the last non-zero digit in a primorial\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46060 \\\"Identify primorial primes\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e*;-)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60947,"title":"Euclid primes","description":"Historical context\r\nEuclid, the greek mathematician proved at his time that the prime numbers set is an infinite. By the way he built some numbers, the Euclid numbers, an infinity of which are thought to be prime (it is still a conjecture). These numbers write themselves as e(n) = 1 + the nth primorial : (notre that 2 and 3 are the two first Euclid primes)\r\n\r\n                                        \r\n \r\n\r\nProblem statement\r\n\r\nFor a given integer m, m \u003e= 2, list the Euclid prime numbers less or equal to m in a vector ep. Also compute their frequency / ratio f compare to all the primes in the same range. If , your function should return ep = [] and f = 0.\r\n \r\n\r\nExamples (check the tests for more) \r\n\r\nIf m = 10 then ep = [2, 3, 7] and f = 3/4 since 5 is the only prime number over the four ones less than 10 which is not an Euclid prime;\r\n \r\nIf m = 300 then ep = [2, 3, 7, 31, 211] and f = 5/62 since 2 and 3 are Euclid primes, 7 = 2x3+1, 31 = 2x3x5+1, 211 = 2x3x5x7+1, and there are in total  62 prime numbers in the range [2; 300] ; \r\n \r\nIf m = 1 then ep = [] and f = 0;\r\n \r\nTip \r\n here.\r\n \r\n\r\nFobidden functions \r\nregexp\r\n \r\nstr2num\r\n \r\nassignin\r\n\r\n\r\nSee also\r\nProblem 60942. Euler primes\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1228.27px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 614.133px; transform-origin: 408px 614.133px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 364.45px 8px; transform-origin: 364.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEuclid, the greek mathematician proved at his time that the prime numbers set is an infinite. By the way he built some numbers, the Euclid numbers, an infinity of which are thought to be prime (it is still a conjecture). These numbers write themselves as e(n) = 1 + the nth primorial : (notre that 2 and 3 are the two first Euclid primes)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 57.8px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 20px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 20px; outline-color: rgb(60, 60, 60); perspective-origin: 385px 28.9px; text-align: left; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 385px 28.9px; white-space-collapse: preserve; margin-left: 4px; margin-top: 20px; margin-bottom: 5px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 111px 11px; transform-origin: 111px 11px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                        \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-28px\"\u003e\u003cimg 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\" width=\"174.5\" height=\"58\" style=\"width: 174.5px; height: 58px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2917px 8px; transform-origin: 60.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 22.1833px 8px; transform-origin: 22.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u0026gt;= 2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 141.2px 8px; transform-origin: 141.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elist the Euclid prime numbers less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.1667px 8px; transform-origin: 36.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 8px; transform-origin: 7.78333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.4583px 8px; transform-origin: 96.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Also compute their frequency / ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ef \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 146.242px 8px; transform-origin: 146.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ecompare to all the primes in the same range. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.8583px 8px; transform-origin: 82.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.6917px 8px; transform-origin: 24.6917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e ep = [] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e f = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3333px 8px; transform-origin: 79.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(check the tests for more)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 10\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 40.25px 8px; transform-origin: 40.25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = [2, 3, 7]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.975px 8px; transform-origin: 21.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 3/4 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.3417px 8px; transform-origin: 16.3417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 220.933px 8px; transform-origin: 220.933px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e5 is the only prime number over the four ones less than 10 which is not an Euclid prime\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.875px 8px; transform-origin: 25.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 300\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 66.9583px 8px; transform-origin: 66.9583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = [2, 3, 7, 31, 211]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.8667px 8px; transform-origin: 25.8667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 5/62 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e3\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 59.9px 8px; transform-origin: 59.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are Euclid primes, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 100.45px 8px; transform-origin: 100.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e7 = 2x3+1, 31 = 2x3x5+1, 211 = 2x3x5x7+1, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 41.625px 8px; transform-origin: 41.625px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand there are\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.6167px 8px; transform-origin: 20.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ein total\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 9.725px 8px; transform-origin: 9.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e62 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85.5833px 8px; transform-origin: 85.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eprime numbers in the range\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 31.1px 8px; transform-origin: 31.1px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e [2; 300] ; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.0333px 8px; transform-origin: 20.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.8083px 8px; transform-origin: 20.8083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14.2px 8px; transform-origin: 14.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 11px; text-align: left; transform-origin: 385px 11px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"49\" height=\"20\" style=\"width: 49px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60942-euler-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60942. Euler primes\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ep,f] = euclid_primes(n)\r\n  ep = n;\r\n  f = 1;\r\nend","test_suite":"%%\r\nm = 5;\r\nep_correct = [2,3];\r\nf_correct = 2/3;\r\n[ep,f] = euclid_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 10;\r\nep_correct = [2,3,7];\r\nf_correct = 3/4;\r\n[ep,f] = euclid_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 100;\r\nep_correct = [2,3,7,31];\r\nf_correct = 4/25;\r\n[ep,f] = euclid_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 300;\r\nep_correct = [2,3,7,31,211];\r\nf_correct = 5/62;\r\n[ep,f] = euclid_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('euclid_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:54:43.000Z","deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":"2025-07-09T05:57:31.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-27T12:58:55.000Z","updated_at":"2026-03-16T20:27:53.000Z","published_at":"2025-06-27T13:49:30.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEuclid, the greek mathematician proved at his time that the prime numbers set is an infinite. By the way he built some numbers, the Euclid numbers, an infinity of which are thought to be prime (it is still a conjecture). These numbers write themselves as e(n) = 1 + the nth primorial : (notre that 2 and 3 are the two first Euclid primes)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"heading\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee(n) = 1 + \\\\prod_{p_n \\\\in \\\\mathbb{P}} p_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u0026gt;= 2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003elist the Euclid prime numbers less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Also compute their frequency / ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ecompare to all the primes in the same range. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em \u0026lt; 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e ep = [] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(check the tests for more)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 10\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = [2, 3, 7]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 3/4 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e5 is the only prime number over the four ones less than 10 which is not an Euclid prime\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 300\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = [2, 3, 7, 31, 211]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 5/62 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e3\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e are Euclid primes, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e7 = 2x3+1, 31 = 2x3x5+1, 211 = 2x3x5x7+1, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand there are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ein total\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e62 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eprime numbers in the range\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e [2; 300] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n \\\\leq 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Euler primes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46060,"title":"Identify primorial primes","description":"Cody Problems \u003chttps://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial 46054\u003e and \u003chttps://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial 46057\u003e deal with the \u003chttps://mathworld.wolfram.com/Primorial.html primorial\u003e p_n#, which is the product of all primes up to and including the nth prime p_n. The primorials corresponding to n = 1 through 5 are 2, 6, 30, 210, and 2310. \u003chttps://mathworld.wolfram.com/PrimorialPrime.html Primorial primes\u003e are prime numbers that are either one smaller or one larger than a primorial. Examples include 3, 5, 7, 29, 31, 211, 2309, and 2311. \r\n\r\nWrite a function to identify primorial primes and their associated primorials. Given an input |x|, return a variable |y| with one of the following values:\r\n\r\n y = 1    if x is a primorial prime\r\n y = 0    if x is prime but not a primorial prime\r\n y = -Inf if x is composite or 1\r\n\r\nAlso return a variable |p| that is the associated primorial in the first case and |NaN| in the other two cases.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 246.8px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 123.4px; transform-origin: 407px 123.4px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; 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perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/Primorial.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eprimorial\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; 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unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 104.25px 7.91667px; transform-origin: 104.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The primorials corresponding to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 131.267px 7.91667px; transform-origin: 131.267px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 1 through 5 are 2, 6, 30, 210, and 2310.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/PrimorialPrime.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrimorial primes\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.6333px 7.91667px; transform-origin: 62.6333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are prime numbers that are either one smaller or one larger than a primorial. Examples include 3, 5, 7, 29, 31, 211, 2309, and 2311.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 278px 7.91667px; transform-origin: 278px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to identify primorial primes and their associated primorials. Given an input\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.91667px; transform-origin: 3.85px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ex\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.4667px 7.91667px; transform-origin: 54.4667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, return a variable\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.91667px; transform-origin: 3.85px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ey\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.7333px 7.91667px; transform-origin: 37.7333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with one of the following values:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 61.3px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 30.65px; transform-origin: 404px 30.65px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 134.75px 7.91667px; transform-origin: 134.75px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 38.5px 7.91667px; transform-origin: 38.5px 7.91667px; \"\u003e y = 1    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); perspective-origin: 11.55px 7.91667px; text-decoration: none; text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); transform-origin: 11.55px 7.91667px; \"\u003eif \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; \"\u003ex is \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 65.45px 7.91667px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 65.45px 7.91667px; \"\u003ea primorial prime\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 188.65px 7.91667px; transform-origin: 188.65px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 38.5px 7.91667px; transform-origin: 38.5px 7.91667px; \"\u003e y = 0    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); perspective-origin: 11.55px 7.91667px; text-decoration: none; text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); transform-origin: 11.55px 7.91667px; \"\u003eif \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; \"\u003ex is \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 119.35px 7.91667px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 119.35px 7.91667px; \"\u003eprime but not a primorial prime\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 123.2px 7.91667px; transform-origin: 123.2px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 38.5px 7.91667px; transform-origin: 38.5px 7.91667px; \"\u003e y = -Inf \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); perspective-origin: 11.55px 7.91667px; text-decoration: none; text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); transform-origin: 11.55px 7.91667px; \"\u003eif \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; \"\u003ex is \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 53.9px 7.91667px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 53.9px 7.91667px; \"\u003ecomposite or 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.1333px 7.91667px; transform-origin: 66.1333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAlso return a variable\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.91667px; transform-origin: 3.85px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 159.083px 7.91667px; transform-origin: 159.083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that is the associated primorial in the first case and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eNaN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.95px 7.91667px; transform-origin: 71.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the other two cases.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1833px 7.91667px; transform-origin: 29.1833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problems 46054 \"Count trailing zeros in a primorial\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.91667px; transform-origin: 15.5667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46057 \"Find the last non-zero digit in a primorial\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [y,p] = isPrimorialPrime(x)\r\n%  x = candidate primorial prime\r\n%  p = primorial (or NaN)\r\n%  y = 1 for primorial primes, 0 for primes that are not primorial, -Inf for 1 or composites\r\n\r\n  y = f(x);\r\n  p = f(x);\r\nend","test_suite":"%%\r\nx = 1;\r\ny_correct = -Inf;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 2;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 3;\r\ny_correct = 1;\r\np_correct = 2;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 5;\r\ny_correct = 1;\r\np_correct = 6;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 7;\r\ny_correct = 1;\r\np_correct = 6;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 13;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 17;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 29;\r\ny_correct = 1;\r\np_correct = 30;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 31;\r\ny_correct = 1;\r\np_correct = 30;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 61;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 128;\r\ny_correct = -Inf;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 211;\r\ny_correct = 1;\r\np_correct = 210;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 599;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 2311;\r\ny_correct = 1;\r\np_correct = 2310;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 30029;\r\ny_correct = 1;\r\np_correct = 30030;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 999983;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 142438007;\r\ny_correct = -Inf;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 200560490131;\r\ny_correct = 1;\r\np_correct = 200560490130;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":35,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-27T02:21:36.000Z","updated_at":"2025-11-16T17:45:44.000Z","published_at":"2020-07-27T04:12:44.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46054\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46057\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deal with the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/Primorial.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eprimorial\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#, which is the product of all primes up to and including the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The primorials corresponding to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\\n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 1 through 5 are 2, 6, 30, 210, and 2310.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/PrimorialPrime.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrimorial primes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e are prime numbers that are either one smaller or one larger than a primorial. Examples include 3, 5, 7, 29, 31, 211, 2309, and 2311.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to identify primorial primes and their associated primorials. Given an input\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, return a variable\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e with one of the following values:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ y = 1    if x is a primorial prime\\n y = 0    if x is prime but not a primorial prime\\n y = -Inf if x is composite or 1]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAlso return a variable\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e that is the associated primorial in the first case and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNaN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in the other two cases.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems 46054 \\\"Count trailing zeros in a primorial\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46057 \\\"Find the last non-zero digit in a primorial\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46087,"title":"Investigate the frequency of last non-zero digits of primorials","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 529.633px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 264.817px; transform-origin: 407px 264.817px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 64.6333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 32.3167px; text-align: left; transform-origin: 384px 32.3167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 49.8px 7.8px; transform-origin: 49.8px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCody Problems \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46054\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.8px; transform-origin: 3.88333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46057\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 7.8px; transform-origin: 17.5px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46060\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.1333px 7.8px; transform-origin: 66.1333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e involve the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAABWklEQVRYhe2Wa62EMBSEPw84wMAaQAEKcLAOcIAFNKwEPGABDVi49wfnZCe9pWVJ9/7qJE029DFzHp0tVFRUVFRUfBkN0AE9MAKzDYAWmIDFxmjri5O/gB8bkxFt9nuWubW0AEyAEywmRrHIfF+afJTDQ+JwfipNvtrBO0et/428IR01RviVtPdy8HCyZpU1sczchnfzTryTW/KZuY0tc7Cm/FGSWKN6nszvNj+WJIajxqmovNZnWWk5PMKFNRxBPLlgRups4WLvhTncJMLdfF4mfhXB2VvhKdXFvRzSZfZ7P4wmBN6ecHZzwJSqcQy2ceD6dfIo1fM9G8m0q2vd6WJtVo3SxSThCvcbxPBu1k2++R9U0oKvWGoO3pAatWcz2StqqbH7fQVuTlrbVb79SX1jxL7RI+/47JHg9V6C716G6KPDXy6x8Qn5gyPFYaP6c6z4a6eiouISfgF1CoWEjhYNVwAAAABJRU5ErkJggg==\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 180.533px 7.8px; transform-origin: 180.533px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#, or the product of prime numbers up to and including the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2333px 7.8px; transform-origin: 27.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 338px 7.8px; transform-origin: 338px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In particular, the second of these deals with the last non-zero digit (LNZD) in the primorial. For example, the first eleven primorials are\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, and 6469693230\" style=\"width: 541px; height: 18px;\" width=\"541\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.8px; transform-origin: 3.88333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 263.7px 7.8px; transform-origin: 263.7px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAfter the second primorial, the LNZD is 1, 3, 7, or 9. This observation holds for larger \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 102.317px 7.8px; transform-origin: 102.317px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as well because prime numbers greater than 5 end in 1, 3, 7, or 9 and products of numbers whose LNZDs are 1, 3, 7, or 9 also have a LNZD of 1, 3, 7, or 9. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.9667px 7.8px; transform-origin: 50.9667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eMathematicians \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ehave determined\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 278.483px 7.8px; transform-origin: 278.483px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that the last digits of prime numbers repeat themselves less often than expected. If these digits occurred randomly, 25% of the primes that follow primes ending in a 1 should also end in a 1. However, as verified in \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45994-investigate-the-frequency-of-last-digits-of-prime-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"perspective-origin: 65.75px 7.8px; transform-origin: 65.75px 7.8px; \"\u003eCody Problem 45994\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.45px 7.8px; transform-origin: 54.45px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, for primes up to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"100 million\" style=\"width: 23px; height: 19px;\" width=\"23\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 123.683px 7.8px; transform-origin: 123.683px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the actual frequency is less than 18%. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 374.2px 7.8px; transform-origin: 374.2px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhat about primorials? What happens when basic multiplication facts interact with the frequency of last digits of primes? Will the frequencies be similarly skewed? Or will they all approach 25%? \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 86.45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 43.2333px; text-align: left; transform-origin: 384px 43.2333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 142.233px 7.8px; transform-origin: 142.233px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes as input an integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.217px 7.8px; transform-origin: 211.217px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and computes the frequency of the last digits of primorials between \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_3\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.8667px 7.8px; transform-origin: 11.8667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# = 30 and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 348.9px 7.8px; transform-origin: 348.9px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#. Return a matrix whose rows correspond to the digits of the first primorial and columns correspond to the digits of the next primorial. Please remember to (a) omit \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAABJklEQVRYhe2WURGDMAyGPw84wMAMoAAFc4CDOsACGiYBD1iYBixsD01uuR7tBpQ95b/LsbuUfE2aZoDL5XK5XBerATqgBwIwiQG0wAjMYkHWV4c/gJfYKKCn/J6Mb6m9AWQDCphlM1az8fe14cEET8Gpf6wNXyTwSjzrv8EbylkjwG9l7wq+rHoT+J5Zs5g1aWVaPg27uyrazSvbndySr0wHDGfgz0xglS35LbNGe2IX3GY1ZPyr+EMhziH4nXJWeta5qpyC28mWnrf2wpS+VAuuJbVXqCdmvBAb6hftht8MeCQeQZDn1qCpCrdTK9fFl8H1z2I9Cd4N/2WkXga3I3Xrfl8CbwSsU00z7zj3kaBHWKyifrls2RF4S6xcMHbktrhcrjp6A2YcfI+YyrElAAAAAElFTkSuQmCC\" alt=\"p_1\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.3167px 7.8px; transform-origin: 31.3167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# = 2 and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_2\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 148.017px 7.8px; transform-origin: 148.017px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# = 6 and (b) account for the primorial following \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.2167px 7.8px; transform-origin: 20.2167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#. For example, if your function is given \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.5px 7.8px; transform-origin: 81.5px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 10, then it should return\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 81.7333px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 40.8667px; transform-origin: 404px 40.8667px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 154px 8.25px; transform-origin: 154px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0.3333    0.3333         0    0.3333\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 154px 8.25px; transform-origin: 154px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0.6667    0.3333         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 154px 8.25px; transform-origin: 154px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e         0    1.0000         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 154px 8.25px; transform-origin: 154px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e         0         0    1.0000         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 31.9167px; text-align: left; transform-origin: 384px 31.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.2833px 7.8px; transform-origin: 39.2833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThat is, with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.683px 7.8px; transform-origin: 110.683px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 10, we consider eight primorials (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAABY0lEQVRYhe2WYZGEMAyFPw84qAEMoAAF6wAHdYAFNJwEPGABDVjY/dFkyfTapb1h71ffTIddWvqSvCQtNDQ0NDQ0fBkdMAAj4IFFBoADZmCV4WX97eQ/wFPGLES7/F7M3Ha3AYgBSrCKMRarmR/vJvdm85g4np/vJt9k44Og9b+Rd3z2GiHMhd0Bkxg41ZKPZuNHZs1m1tjIPDjzZOdMylT0ktBsPkhnsiMdGSeElkirpjgCanFJyHvzfor+w1k1RXlhvUpZ6wgReRI0vYJKGBuVhGqW+0C1zkUlxkpFyG1ni/XWXFjijxJwQnxQUYoaUltCI8HjjaDhFXqCJLYLXkrUm8UzQQIvz+JSiaAy7lcLbdcqSpBCqFwfDyAN03EjMQSncj0DKGupf4XeAbKwLbW6HwsGfrfjjuB18tjtZEK7mno+UH9JUNk2MWKUvXLnw/vmkhq15B3naeYpK8uGhobv4gUDlYEOU1DyDQAAAABJRU5ErkJggg==\" alt=\"p_3\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.5167px 7.8px; transform-origin: 31.5167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# through \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_10\" style=\"width: 20.5px; height: 20px;\" width=\"20.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 169.567px 7.8px; transform-origin: 169.567px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#). Of these eight, three have a LNZD of 3 (30, 30030, and 6469693230). Two of those (i.e., 66.7%) are followed by primorials with a LNZD of 1, and one (33.3%) is followed by a primorial with a LNZD of 3.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = primorialLastNZDigitFreq(n)\r\n  f = ...;\r\nend","test_suite":"%%\r\nn = 10;\r\nf_correct = [0.3333 0.3333 0 0.3333; 0.6667 0.3333 0 0; 0 1.0000 0 0; 0 0 1.0000 0];\r\nassert(isequal(round(primorialLastNZDigitFreq(n),4),f_correct))\r\n\r\n%%\r\nn = 100;\r\nf_correct = [0.1923 0.2308 0.3077 0.2692; 0.3636 0.2273 0.1818 0.2273; 0.2800 0.2800 0.2400 0.2000; 0.2400 0.1600 0.2800 0.3200];\r\nassert(isequal(round(primorialLastNZDigitFreq(n),4),f_correct))\r\n\r\n%%\r\nn = 1000;\r\nf_correct = [0.2394 0.2625 0.2432 0.2548; 0.3095 0.2460 0.2143 0.2302; 0.2389 0.2672 0.2551 0.2389; 0.2500 0.2292 0.2792 0.2417];\r\nassert(isequal(round(primorialLastNZDigitFreq(n),4),f_correct))\r\n\r\n%%\r\nn = 1e4;\r\nf_correct = [0.2426 0.2512 0.2471 0.2590; 0.2551 0.2544 0.2385 0.2520; 0.2374 0.2602 0.2466 0.2558; 0.2394 0.2453 0.2654 0.2500];\r\nassert(isequal(round(primorialLastNZDigitFreq(n),4),f_correct))\r\n\r\n%%\r\nn = 1e6;\r\nf_correct = [0.2489 0.2493 0.2509 0.2509; 0.2505 0.2506 0.2484 0.2505; 0.2499 0.2505 0.2504 0.2492; 0.2500 0.2494 0.2507 0.2498];\r\nassert(isequal(round(primorialLastNZDigitFreq(n),4),f_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":14,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-08-01T14:57:24.000Z","updated_at":"2025-11-15T15:20:56.000Z","published_at":"2020-08-01T16:58:23.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46054\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46057\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46060\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e involve the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#, or the product of prime numbers up to and including the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In particular, the second of these deals with the last non-zero digit (LNZD) in the primorial. For example, the first eleven primorials are\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, and 6469693230\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, {\\\\rm and}\\\\,\\\\,200560490130 \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAfter the second primorial, the LNZD is 1, 3, 7, or 9. This observation holds for larger \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as well because prime numbers greater than 5 end in 1, 3, 7, or 9 and products of numbers whose LNZDs are 1, 3, 7, or 9 also have a LNZD of 1, 3, 7, or 9. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMathematicians \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehave determined\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e that the last digits of prime numbers repeat themselves less often than expected. If these digits occurred randomly, 25% of the primes that follow primes ending in a 1 should also end in a 1. However, as verified in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45994-investigate-the-frequency-of-last-digits-of-prime-numbers\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45994\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, for primes up to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"100 million\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e10^8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the actual frequency is less than 18%. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat about primorials? What happens when basic multiplication facts interact with the frequency of last digits of primes? Will the frequencies be similarly skewed? Or will they all approach 25%? \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes as input an integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and computes the frequency of the last digits of primorials between \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# = 30 and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#. Return a matrix whose rows correspond to the digits of the first primorial and columns correspond to the digits of the next primorial. Please remember to (a) omit \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# = 2 and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# = 6 and (b) account for the primorial following \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#. For example, if your function is given \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 10, then it should return\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    0.3333    0.3333         0    0.3333\\n    0.6667    0.3333         0         0\\n         0    1.0000         0         0\\n         0         0    1.0000         0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThat is, with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 10, we consider eight primorials (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# through \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_10\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_{10}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#). Of these eight, three have a LNZD of 3 (30, 30030, and 6469693230). Two of those (i.e., 66.7%) are followed by primorials with a LNZD of 1, and one (33.3%) is followed by a primorial with a LNZD of 3.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46057,"title":"Find the last non-zero digit in a primorial","description":"\u003chttps://www.mathworks.com/matlabcentral/cody/groups/2001/problems/45251 Cody Problem 45251\u003e asked us to find the last non-zero digit in a factorial. For this problem consider the primorial p_n#, the product of primes up to and including the nth prime p_n. For example, for n = 12, the primorial is 7420738134810, whose last non-zero digit is 1.\r\n\r\nFind the last non-zero digit in the primorial p_n#. \r\n\r\nSee also \u003chttps://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial Cody Problem 46054\u003e \"Count trailing zeros in a primorial\".","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 123.75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 61.8833px; transform-origin: 407px 61.8833px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.75px; text-align: left; transform-origin: 384px 31.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/groups/2001/problems/45251\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 45251\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 284.733px 7.91667px; transform-origin: 284.733px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asked us to find the last non-zero digit in a factorial. For this problem consider the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.45px 7.91667px; transform-origin: 19.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#, the product of primes up to and including the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2333px 7.91667px; transform-origin: 27.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 56.7833px 7.91667px; transform-origin: 56.7833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example, for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.75px 7.91667px; transform-origin: 143.75px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 12, the primorial is 7420738134810, whose last non-zero digit is 1.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.6333px; text-align: left; transform-origin: 384px 10.6333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.267px 7.91667px; transform-origin: 132.267px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFind the last non-zero digit in the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 7.91667px; transform-origin: 5.83333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2333px 7.91667px; transform-origin: 27.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problems 46054\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.783px 7.91667px; transform-origin: 124.783px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \"Count trailing zeros in a primorial\" and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46060 \"Identify primorial primes\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = primorialLastNonzeroDigit(n)\r\n  y = f(n);\r\nend","test_suite":"%%\r\nn = 1;\r\ny_correct = 2;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 2;\r\ny_correct = 6;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 3;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 4;\r\ny_correct = 1;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 12;\r\ny_correct = 1;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 29;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 53;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 89;\r\ny_correct = 7;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 163;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 211;\r\ny_correct = 1;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 254;\r\ny_correct = 9;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 350;\r\ny_correct = 7;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 1379;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 37913;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 1379137;\r\ny_correct = 9;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":31,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-26T18:50:12.000Z","updated_at":"2025-11-15T15:07:03.000Z","published_at":"2020-07-26T19:43:27.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/groups/2001/problems/45251\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45251\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asked us to find the last non-zero digit in a factorial. For this problem consider the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#, the product of primes up to and including the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example, for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 12, the primorial is 7420738134810, whose last non-zero digit is 1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the last non-zero digit in the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems 46054\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \\\"Count trailing zeros in a primorial\\\" and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46060 \\\"Identify primorial primes\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1688,"title":"Prime Sequences: AP-k Minimum Final Value","description":"Welcome to Fun with Primes. Today we will find the Minimum Final Value AP-k sequences for n_max=3:12 given the primorial and knowledge that the solution is of the form a + b * k# * n.\r\n\r\nThe AP-k of n sequence is n_max+1 primes of the form a + b * k# * n where n=0:n_max. The value of \"a\" is a prime and k# is the primorial.\r\n\r\nThe primorial k# is the product of all primes ≤ k, e.g. 10# = 2 · 3 · 5 · 7.\r\n\r\n*Input:* (k, n_max) \r\n\r\n*Output:* [a, b] for the equation Prime = a + b * k# * n,  n=0:n_max; Prime(n_max) must be the optimum minimum.\r\n\r\n*Value Range Limits:* [a\u003c150,000 , b\u003c8 ]\r\n\r\n*Example:* \r\n\r\n(13, 13) yields [31385539,14 ]; 31385539 + 14·13#·n (End Prime 36850999) \r\n\r\n*Commentary:*\r\n\r\n(13, 16) has a non-minimal end [17, 11387819007325752 ] to give Primes=17 + 11387819007325752·13#·n\r\n\r\nThe current June 2013 record for n is 25 via PrimeGrid:  43142746595714191 + 23681770·23#·n\r\n\r\n\r\n\r\n","description_html":"\u003cp\u003eWelcome to Fun with Primes. Today we will find the Minimum Final Value AP-k sequences for n_max=3:12 given the primorial and knowledge that the solution is of the form a + b * k# * n.\u003c/p\u003e\u003cp\u003eThe AP-k of n sequence is n_max+1 primes of the form a + b * k# * n where n=0:n_max. The value of \"a\" is a prime and k# is the primorial.\u003c/p\u003e\u003cp\u003eThe primorial k# is the product of all primes ≤ k, e.g. 10# = 2 · 3 · 5 · 7.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e (k, n_max)\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e [a, b] for the equation Prime = a + b * k# * n,  n=0:n_max; Prime(n_max) must be the optimum minimum.\u003c/p\u003e\u003cp\u003e\u003cb\u003eValue Range Limits:\u003c/b\u003e [a\u0026lt;150,000 , b\u0026lt;8 ]\u003c/p\u003e\u003cp\u003e\u003cb\u003eExample:\u003c/b\u003e\u003c/p\u003e\u003cp\u003e(13, 13) yields [31385539,14 ]; 31385539 + 14·13#·n (End Prime 36850999)\u003c/p\u003e\u003cp\u003e\u003cb\u003eCommentary:\u003c/b\u003e\u003c/p\u003e\u003cp\u003e(13, 16) has a non-minimal end [17, 11387819007325752 ] to give Primes=17 + 11387819007325752·13#·n\u003c/p\u003e\u003cp\u003eThe current June 2013 record for n is 25 via PrimeGrid:  43142746595714191 + 23681770·23#·n\u003c/p\u003e","function_template":"function [a,b]=APk_min_end(k,n)\r\n% a+b*k#*(0:n) are all primes\r\n a=0;\r\n b=0;\r\nend\r\n","test_suite":"tic\r\nn=3; p=3;\r\n[a,b]=APk_min_end(p,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(p))*n,23))\r\n%%\r\nn=4; p=3;\r\n[a,b]=APk_min_end(p,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(p))*n,29))\r\n%%\r\nn=5; p=5;\r\n[a,b]=APk_min_end(p,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(p))*n,157))\r\n%%\r\nn=6; k=5;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,907))\r\n%%\r\nn=7; k=7;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,1669))\r\n%%\r\nn=8; k=7;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,1879))\r\n%%\r\nn=9; k=7;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,2089))\r\n%%\r\nn=10; k=11;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,249037))\r\n%%\r\nn=11; k=11;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,262897))\r\n%%\r\nn=12; k=13;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,725663))","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":"2013-06-30T03:36:26.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-06-30T03:04:32.000Z","updated_at":"2013-06-30T04:28:44.000Z","published_at":"2013-06-30T03:36:26.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWelcome to Fun with Primes. Today we will find the Minimum Final Value AP-k sequences for n_max=3:12 given the primorial and knowledge that the solution is of the form a + b * k# * n.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe AP-k of n sequence is n_max+1 primes of the form a + b * k# * n where n=0:n_max. The value of \\\"a\\\" is a prime and k# is the primorial.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe primorial k# is the product of all primes ≤ k, e.g. 10# = 2 · 3 · 5 · 7.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (k, n_max)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e [a, b] for the equation Prime = a + b * k# * n, n=0:n_max; Prime(n_max) must be the optimum minimum.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eValue Range Limits:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e [a\u0026lt;150,000 , b\u0026lt;8 ]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(13, 13) yields [31385539,14 ]; 31385539 + 14·13#·n (End Prime 36850999)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCommentary:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(13, 16) has a non-minimal end [17, 11387819007325752 ] to give Primes=17 + 11387819007325752·13#·n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe current June 2013 record for n is 25 via PrimeGrid: 43142746595714191 + 23681770·23#·n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":46054,"title":"Count trailing zeros in a primorial","description":"\u003chttps://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial Cody Problem 44068\u003e asked us to count the  trailing zeros in a factorial. This problem deals with the  _\u003chttps://mathworld.wolfram.com/Primorial.html primorial\u003e_. If p_n is the nth prime number, then the primorial p_n# is the product of the prime numbers up to and including p_n. For example, if n = 5, then p_5# = 2*3*5*7*11 = 2310, which has one trailing zero. \r\n\r\nCount the trailing zeros in the primorial p_n#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.* \r\n\r\nPlease make your code general because I might add more tests later. \r\n\r\n*;-)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 205px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 102.5px; transform-origin: 407px 102.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.75px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.8833px; text-align: left; transform-origin: 384px 31.8833px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 44068\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 238.833px 7.91667px; transform-origin: 238.833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asked us to count the trailing zeros in a factorial. This problem deals with the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/Primorial.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eprimorial\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 7.91667px; transform-origin: 9.71667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6167px 7.91667px; transform-origin: 20.6167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.28333px 7.91667px; transform-origin: 7.28333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime number, then the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 178.55px 7.91667px; transform-origin: 178.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# is the product of the prime numbers up to and including \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 52.1167px 7.91667px; transform-origin: 52.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example, if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.85px 7.91667px; transform-origin: 29.85px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 5, then \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_5\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.8667px 7.91667px; transform-origin: 11.8667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# = \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2*3*5*7*11\" style=\"width: 92px; height: 18px;\" width=\"92\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 112.233px 7.91667px; transform-origin: 112.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 2310, which has one trailing zero.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.1333px; text-align: left; transform-origin: 384px 21.1333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 121.767px 7.91667px; transform-origin: 121.767px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCount the trailing zeros in the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 254.467px 7.91667px; transform-origin: 254.467px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.*\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 215.1px 7.91667px; transform-origin: 215.1px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ePlease make your code general because I might add more tests later.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1833px 7.91667px; transform-origin: 29.1833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problems 46057 \"Find the last non-zero digit in a primorial\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.91667px; transform-origin: 15.5667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46060 \"Identify primorial primes\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.33333px 7.91667px; transform-origin: 9.33333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e*;-)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = primorialTrailingZeros(n)\r\n  y = f(n);\r\nend","test_suite":"%%\r\nn = 1;\r\ny_correct = 0;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 2;\r\ny_correct = 0;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 5;\r\ny_correct = 1;\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 10;\r\ny_correct = sin((n+3)*pi/2);\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 100;\r\ny_correct = sum(1./2.^(1:n));\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 1e4;\r\ny_correct = besselj(mod(n,10),prod(num2str(n)'-'0'));\r\nassert(isequal(primorialTrailingZeros(n),y_correct))\r\n\r\n%%\r\nn = 1e6;\r\nf = [factor(log10(n)) 3 7 7 89];\r\ny_correct = length(primes(n))/prod(f);\r\nassert(isequal(primorialTrailingZeros(n),y_correct))","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":83,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-26T15:54:06.000Z","updated_at":"2025-12-09T22:52:30.000Z","published_at":"2020-07-26T16:20:27.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/44068-the-number-of-trailing-zero-digit-of-a-factorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 44068\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asked us to count the trailing zeros in a factorial. This problem deals with the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/Primorial.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eprimorial\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime number, then the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# is the product of the prime numbers up to and including \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example, if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 5, then \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# = \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2*3*5*7*11\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2\\\\cdot3\\\\cdot5\\\\cdot7\\\\cdot11\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 2310, which has one trailing zero.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCount the trailing zeros in the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease make your code general because I might add more tests later.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems 46057 \\\"Find the last non-zero digit in a primorial\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46060 \\\"Identify primorial primes\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e*;-)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60947,"title":"Euclid primes","description":"Historical context\r\nEuclid, the greek mathematician proved at his time that the prime numbers set is an infinite. By the way he built some numbers, the Euclid numbers, an infinity of which are thought to be prime (it is still a conjecture). These numbers write themselves as e(n) = 1 + the nth primorial : (notre that 2 and 3 are the two first Euclid primes)\r\n\r\n                                        \r\n \r\n\r\nProblem statement\r\n\r\nFor a given integer m, m \u003e= 2, list the Euclid prime numbers less or equal to m in a vector ep. Also compute their frequency / ratio f compare to all the primes in the same range. If , your function should return ep = [] and f = 0.\r\n \r\n\r\nExamples (check the tests for more) \r\n\r\nIf m = 10 then ep = [2, 3, 7] and f = 3/4 since 5 is the only prime number over the four ones less than 10 which is not an Euclid prime;\r\n \r\nIf m = 300 then ep = [2, 3, 7, 31, 211] and f = 5/62 since 2 and 3 are Euclid primes, 7 = 2x3+1, 31 = 2x3x5+1, 211 = 2x3x5x7+1, and there are in total  62 prime numbers in the range [2; 300] ; \r\n \r\nIf m = 1 then ep = [] and f = 0;\r\n \r\nTip \r\n here.\r\n \r\n\r\nFobidden functions \r\nregexp\r\n \r\nstr2num\r\n \r\nassignin\r\n\r\n\r\nSee also\r\nProblem 60942. Euler primes\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1228.27px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 614.133px; transform-origin: 408px 614.133px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 364.45px 8px; transform-origin: 364.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEuclid, the greek mathematician proved at his time that the prime numbers set is an infinite. By the way he built some numbers, the Euclid numbers, an infinity of which are thought to be prime (it is still a conjecture). These numbers write themselves as e(n) = 1 + the nth primorial : (notre that 2 and 3 are the two first Euclid primes)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan 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\" width=\"174.5\" height=\"58\" style=\"width: 174.5px; height: 58px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2917px 8px; transform-origin: 60.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.225px 8px; transform-origin: 6.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 22.1833px 8px; transform-origin: 22.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u0026gt;= 2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 141.2px 8px; transform-origin: 141.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elist the Euclid prime numbers less or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.1667px 8px; transform-origin: 36.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 8px; transform-origin: 7.78333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.4583px 8px; transform-origin: 96.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Also compute their frequency / ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ef \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 146.242px 8px; transform-origin: 146.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ecompare to all the primes in the same range. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.8583px 8px; transform-origin: 82.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.6917px 8px; transform-origin: 24.6917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e ep = [] \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e f = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3333px 8px; transform-origin: 79.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(check the tests for more)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 10\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 40.25px 8px; transform-origin: 40.25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = [2, 3, 7]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.975px 8px; transform-origin: 21.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 3/4 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.3417px 8px; transform-origin: 16.3417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 220.933px 8px; transform-origin: 220.933px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e5 is the only prime number over the four ones less than 10 which is not an Euclid prime\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.875px 8px; transform-origin: 25.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 300\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 66.9583px 8px; transform-origin: 66.9583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = [2, 3, 7, 31, 211]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.8667px 8px; transform-origin: 25.8667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 5/62 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e3\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 59.9px 8px; transform-origin: 59.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are Euclid primes, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 100.45px 8px; transform-origin: 100.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e7 = 2x3+1, 31 = 2x3x5+1, 211 = 2x3x5x7+1, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 41.625px 8px; transform-origin: 41.625px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand there are\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.6167px 8px; transform-origin: 20.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ein total\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 9.725px 8px; transform-origin: 9.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e62 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85.5833px 8px; transform-origin: 85.5833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eprime numbers in the range\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 31.1px 8px; transform-origin: 31.1px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e [2; 300] ; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.0333px 8px; transform-origin: 20.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.8083px 8px; transform-origin: 20.8083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eep = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14.2px 8px; transform-origin: 14.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ef = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 11px; text-align: left; transform-origin: 385px 11px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"49\" height=\"20\" style=\"width: 49px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60942-euler-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60942. Euler primes\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ep,f] = euclid_primes(n)\r\n  ep = n;\r\n  f = 1;\r\nend","test_suite":"%%\r\nm = 5;\r\nep_correct = [2,3];\r\nf_correct = 2/3;\r\n[ep,f] = euclid_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 10;\r\nep_correct = [2,3,7];\r\nf_correct = 3/4;\r\n[ep,f] = euclid_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 100;\r\nep_correct = [2,3,7,31];\r\nf_correct = 4/25;\r\n[ep,f] = euclid_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%%\r\nm = 300;\r\nep_correct = [2,3,7,31,211];\r\nf_correct = 5/62;\r\n[ep,f] = euclid_primes(m);\r\nassert(isequal([ep,f],[ep_correct,f_correct]));\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('euclid_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:54:43.000Z","deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":"2025-07-09T05:57:31.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-27T12:58:55.000Z","updated_at":"2026-03-16T20:27:53.000Z","published_at":"2025-06-27T13:49:30.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEuclid, the greek mathematician proved at his time that the prime numbers set is an infinite. By the way he built some numbers, the Euclid numbers, an infinity of which are thought to be prime (it is still a conjecture). These numbers write themselves as e(n) = 1 + the nth primorial : (notre that 2 and 3 are the two first Euclid primes)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"heading\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee(n) = 1 + \\\\prod_{p_n \\\\in \\\\mathbb{P}} p_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u0026gt;= 2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003elist the Euclid prime numbers less or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Also compute their frequency / ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ecompare to all the primes in the same range. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em \u0026lt; 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e ep = [] \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e f = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(check the tests for more)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 10\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = [2, 3, 7]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 3/4 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e5 is the only prime number over the four ones less than 10 which is not an Euclid prime\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 300\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = [2, 3, 7, 31, 211]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 5/62 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e3\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e are Euclid primes, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e7 = 2x3+1, 31 = 2x3x5+1, 211 = 2x3x5x7+1, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand there are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ein total\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e62 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eprime numbers in the range\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e [2; 300] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eep = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n \\\\leq 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Euler primes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46060,"title":"Identify primorial primes","description":"Cody Problems \u003chttps://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial 46054\u003e and \u003chttps://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial 46057\u003e deal with the \u003chttps://mathworld.wolfram.com/Primorial.html primorial\u003e p_n#, which is the product of all primes up to and including the nth prime p_n. The primorials corresponding to n = 1 through 5 are 2, 6, 30, 210, and 2310. \u003chttps://mathworld.wolfram.com/PrimorialPrime.html Primorial primes\u003e are prime numbers that are either one smaller or one larger than a primorial. Examples include 3, 5, 7, 29, 31, 211, 2309, and 2311. \r\n\r\nWrite a function to identify primorial primes and their associated primorials. Given an input |x|, return a variable |y| with one of the following values:\r\n\r\n y = 1    if x is a primorial prime\r\n y = 0    if x is prime but not a primorial prime\r\n y = -Inf if x is composite or 1\r\n\r\nAlso return a variable |p| that is the associated primorial in the first case and |NaN| in the other two cases.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 246.8px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 123.4px; transform-origin: 407px 123.4px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; 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perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/Primorial.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eprimorial\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; 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unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 104.25px 7.91667px; transform-origin: 104.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The primorials corresponding to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 131.267px 7.91667px; transform-origin: 131.267px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 1 through 5 are 2, 6, 30, 210, and 2310.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/PrimorialPrime.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrimorial primes\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.6333px 7.91667px; transform-origin: 62.6333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are prime numbers that are either one smaller or one larger than a primorial. Examples include 3, 5, 7, 29, 31, 211, 2309, and 2311.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 278px 7.91667px; transform-origin: 278px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to identify primorial primes and their associated primorials. Given an input\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.91667px; transform-origin: 3.85px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ex\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.4667px 7.91667px; transform-origin: 54.4667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, return a variable\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.91667px; transform-origin: 3.85px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ey\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.7333px 7.91667px; transform-origin: 37.7333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with one of the following values:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 61.3px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 30.65px; transform-origin: 404px 30.65px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 134.75px 7.91667px; transform-origin: 134.75px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 38.5px 7.91667px; transform-origin: 38.5px 7.91667px; \"\u003e y = 1    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); perspective-origin: 11.55px 7.91667px; text-decoration: none; text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); transform-origin: 11.55px 7.91667px; \"\u003eif \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; \"\u003ex is \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 65.45px 7.91667px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 65.45px 7.91667px; \"\u003ea primorial prime\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 188.65px 7.91667px; transform-origin: 188.65px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 38.5px 7.91667px; transform-origin: 38.5px 7.91667px; \"\u003e y = 0    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); perspective-origin: 11.55px 7.91667px; text-decoration: none; text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); transform-origin: 11.55px 7.91667px; \"\u003eif \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; \"\u003ex is \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 119.35px 7.91667px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 119.35px 7.91667px; \"\u003eprime but not a primorial prime\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 123.2px 7.91667px; transform-origin: 123.2px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 38.5px 7.91667px; transform-origin: 38.5px 7.91667px; \"\u003e y = -Inf \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); perspective-origin: 11.55px 7.91667px; text-decoration: none; text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); transform-origin: 11.55px 7.91667px; \"\u003eif \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; \"\u003ex is \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 53.9px 7.91667px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 53.9px 7.91667px; \"\u003ecomposite or 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.1333px 7.91667px; transform-origin: 66.1333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAlso return a variable\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.91667px; transform-origin: 3.85px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 159.083px 7.91667px; transform-origin: 159.083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that is the associated primorial in the first case and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eNaN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.95px 7.91667px; transform-origin: 71.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the other two cases.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1833px 7.91667px; transform-origin: 29.1833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problems 46054 \"Count trailing zeros in a primorial\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.91667px; transform-origin: 15.5667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46057 \"Find the last non-zero digit in a primorial\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [y,p] = isPrimorialPrime(x)\r\n%  x = candidate primorial prime\r\n%  p = primorial (or NaN)\r\n%  y = 1 for primorial primes, 0 for primes that are not primorial, -Inf for 1 or composites\r\n\r\n  y = f(x);\r\n  p = f(x);\r\nend","test_suite":"%%\r\nx = 1;\r\ny_correct = -Inf;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 2;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 3;\r\ny_correct = 1;\r\np_correct = 2;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 5;\r\ny_correct = 1;\r\np_correct = 6;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 7;\r\ny_correct = 1;\r\np_correct = 6;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 13;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 17;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 29;\r\ny_correct = 1;\r\np_correct = 30;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 31;\r\ny_correct = 1;\r\np_correct = 30;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 61;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 128;\r\ny_correct = -Inf;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 211;\r\ny_correct = 1;\r\np_correct = 210;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 599;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 2311;\r\ny_correct = 1;\r\np_correct = 2310;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 30029;\r\ny_correct = 1;\r\np_correct = 30030;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))\r\n\r\n%%\r\nx = 999983;\r\ny_correct = 0;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 142438007;\r\ny_correct = -Inf;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isnan(p))\r\n\r\n%%\r\nx = 200560490131;\r\ny_correct = 1;\r\np_correct = 200560490130;\r\n[y,p] = isPrimorialPrime(x);\r\nassert(isequal(y,y_correct) \u0026\u0026 isequal(p,p_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":35,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-27T02:21:36.000Z","updated_at":"2025-11-16T17:45:44.000Z","published_at":"2020-07-27T04:12:44.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46054\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46057\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deal with the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/Primorial.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eprimorial\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#, which is the product of all primes up to and including the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The primorials corresponding to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\\n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 1 through 5 are 2, 6, 30, 210, and 2310.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/PrimorialPrime.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrimorial primes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e are prime numbers that are either one smaller or one larger than a primorial. Examples include 3, 5, 7, 29, 31, 211, 2309, and 2311.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to identify primorial primes and their associated primorials. Given an input\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, return a variable\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e with one of the following values:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ y = 1    if x is a primorial prime\\n y = 0    if x is prime but not a primorial prime\\n y = -Inf if x is composite or 1]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAlso return a variable\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e that is the associated primorial in the first case and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNaN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in the other two cases.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems 46054 \\\"Count trailing zeros in a primorial\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46057 \\\"Find the last non-zero digit in a primorial\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46087,"title":"Investigate the frequency of last non-zero digits of primorials","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 529.633px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 264.817px; transform-origin: 407px 264.817px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 64.6333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 32.3167px; text-align: left; transform-origin: 384px 32.3167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 49.8px 7.8px; transform-origin: 49.8px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCody Problems \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46054\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.8px; transform-origin: 3.88333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46057\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 7.8px; transform-origin: 17.5px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46060\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.1333px 7.8px; transform-origin: 66.1333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e involve the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 180.533px 7.8px; transform-origin: 180.533px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#, or the product of prime numbers up to and including the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2333px 7.8px; transform-origin: 27.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 338px 7.8px; transform-origin: 338px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In particular, the second of these deals with the last non-zero digit (LNZD) in the primorial. For example, the first eleven primorials are\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, and 6469693230\" style=\"width: 541px; height: 18px;\" width=\"541\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.8px; transform-origin: 3.88333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 263.7px 7.8px; transform-origin: 263.7px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAfter the second primorial, the LNZD is 1, 3, 7, or 9. This observation holds for larger \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 102.317px 7.8px; transform-origin: 102.317px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as well because prime numbers greater than 5 end in 1, 3, 7, or 9 and products of numbers whose LNZDs are 1, 3, 7, or 9 also have a LNZD of 1, 3, 7, or 9. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.9667px 7.8px; transform-origin: 50.9667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eMathematicians \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ehave determined\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 278.483px 7.8px; transform-origin: 278.483px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that the last digits of prime numbers repeat themselves less often than expected. If these digits occurred randomly, 25% of the primes that follow primes ending in a 1 should also end in a 1. However, as verified in \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45994-investigate-the-frequency-of-last-digits-of-prime-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"perspective-origin: 65.75px 7.8px; transform-origin: 65.75px 7.8px; \"\u003eCody Problem 45994\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.45px 7.8px; transform-origin: 54.45px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, for primes up to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC4AAAAmCAYAAAC76qlaAAABxklEQVRYhe2YbZGEMAyGHw84qAEMrIJVgAMc4AALaEBCPWBhNWBh7wfNNdflI8CyOzfTZ4Yf5NLmbRvS7EEmk/kGt/C4bwux0gAeuIdnAFqg+KaoLSpgTGxFsLWfl2PHM+2w1b4Lx/5jc8ScXcMz7W6a1w+g3xnzT/AOeAKlcUylgjZh/AjUzC++CvMPRPH1zpi/FEqwPJZJZEyamyLEMy9exo1MC9aLMHNn2in5uq3CRdxjQZxnflFCr2INhnirNNiESxVYEyYpMTeX/E3STHZ/6/s4LbxWfvcFH6d89OLKYGvCe0E8nYGDtdwqXKfUWm7KqTyUTfJbjyuIO39o163CxSe9SFI8rwsUW7qzEnvpBFexCC85JlwEtczvrNgP9S0W4TeOCa+DrWBKtV6JLIm1/xB7hW/ddHo+LaoI713waTlRUdJAFuF+Y76WeeFv5xOpcglXVpVTN+MWVuH6tlvDK79Lf+VYhXdGP92LXIpVuM7zasFH1/tL8xvswiGmwVJlkYqy1D2+lT3CHbEXSX1193iqPltwxI9Od29ryI03EsU7YhN2qO+w4piO1c88Hdv5KbdgH8b04f3f/K8kk8lkMpkXfgBUBNSjr8Pw0QAAAABJRU5ErkJggg==\" alt=\"100 million\" style=\"width: 23px; height: 19px;\" width=\"23\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 123.683px 7.8px; transform-origin: 123.683px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the actual frequency is less than 18%. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 374.2px 7.8px; transform-origin: 374.2px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhat about primorials? What happens when basic multiplication facts interact with the frequency of last digits of primes? Will the frequencies be similarly skewed? Or will they all approach 25%? \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 86.45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 43.2333px; text-align: left; transform-origin: 384px 43.2333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 142.233px 7.8px; transform-origin: 142.233px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes as input an integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.217px 7.8px; transform-origin: 211.217px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and computes the frequency of the last digits of primorials between \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_3\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.8667px 7.8px; transform-origin: 11.8667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# = 30 and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 348.9px 7.8px; transform-origin: 348.9px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#. Return a matrix whose rows correspond to the digits of the first primorial and columns correspond to the digits of the next primorial. Please remember to (a) omit \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAABJklEQVRYhe2WURGDMAyGPw84wMAMoAAFc4CDOsACGiYBD1iYBixsD01uuR7tBpQ95b/LsbuUfE2aZoDL5XK5XBerATqgBwIwiQG0wAjMYkHWV4c/gJfYKKCn/J6Mb6m9AWQDCphlM1az8fe14cEET8Gpf6wNXyTwSjzrv8EbylkjwG9l7wq+rHoT+J5Zs5g1aWVaPg27uyrazSvbndySr0wHDGfgz0xglS35LbNGe2IX3GY1ZPyr+EMhziH4nXJWeta5qpyC28mWnrf2wpS+VAuuJbVXqCdmvBAb6hftht8MeCQeQZDn1qCpCrdTK9fFl8H1z2I9Cd4N/2WkXga3I3Xrfl8CbwSsU00z7zj3kaBHWKyifrls2RF4S6xcMHbktrhcrjp6A2YcfI+YyrElAAAAAElFTkSuQmCC\" alt=\"p_1\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.3167px 7.8px; transform-origin: 31.3167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# = 2 and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_2\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 148.017px 7.8px; transform-origin: 148.017px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# = 6 and (b) account for the primorial following \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.2167px 7.8px; transform-origin: 20.2167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#. For example, if your function is given \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.5px 7.8px; transform-origin: 81.5px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 10, then it should return\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 81.7333px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 40.8667px; transform-origin: 404px 40.8667px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 154px 8.25px; transform-origin: 154px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0.3333    0.3333         0    0.3333\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 154px 8.25px; transform-origin: 154px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0.6667    0.3333         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 154px 8.25px; transform-origin: 154px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e         0    1.0000         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 154px 8.25px; transform-origin: 154px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e         0         0    1.0000         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 31.9167px; text-align: left; transform-origin: 384px 31.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.2833px 7.8px; transform-origin: 39.2833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThat is, with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.683px 7.8px; transform-origin: 110.683px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 10, we consider eight primorials (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_3\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.5167px 7.8px; transform-origin: 31.5167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e# through \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_10\" style=\"width: 20.5px; height: 20px;\" width=\"20.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 169.567px 7.8px; transform-origin: 169.567px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#). Of these eight, three have a LNZD of 3 (30, 30030, and 6469693230). Two of those (i.e., 66.7%) are followed by primorials with a LNZD of 1, and one (33.3%) is followed by a primorial with a LNZD of 3.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = primorialLastNZDigitFreq(n)\r\n  f = ...;\r\nend","test_suite":"%%\r\nn = 10;\r\nf_correct = [0.3333 0.3333 0 0.3333; 0.6667 0.3333 0 0; 0 1.0000 0 0; 0 0 1.0000 0];\r\nassert(isequal(round(primorialLastNZDigitFreq(n),4),f_correct))\r\n\r\n%%\r\nn = 100;\r\nf_correct = [0.1923 0.2308 0.3077 0.2692; 0.3636 0.2273 0.1818 0.2273; 0.2800 0.2800 0.2400 0.2000; 0.2400 0.1600 0.2800 0.3200];\r\nassert(isequal(round(primorialLastNZDigitFreq(n),4),f_correct))\r\n\r\n%%\r\nn = 1000;\r\nf_correct = [0.2394 0.2625 0.2432 0.2548; 0.3095 0.2460 0.2143 0.2302; 0.2389 0.2672 0.2551 0.2389; 0.2500 0.2292 0.2792 0.2417];\r\nassert(isequal(round(primorialLastNZDigitFreq(n),4),f_correct))\r\n\r\n%%\r\nn = 1e4;\r\nf_correct = [0.2426 0.2512 0.2471 0.2590; 0.2551 0.2544 0.2385 0.2520; 0.2374 0.2602 0.2466 0.2558; 0.2394 0.2453 0.2654 0.2500];\r\nassert(isequal(round(primorialLastNZDigitFreq(n),4),f_correct))\r\n\r\n%%\r\nn = 1e6;\r\nf_correct = [0.2489 0.2493 0.2509 0.2509; 0.2505 0.2506 0.2484 0.2505; 0.2499 0.2505 0.2504 0.2492; 0.2500 0.2494 0.2507 0.2498];\r\nassert(isequal(round(primorialLastNZDigitFreq(n),4),f_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":14,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-08-01T14:57:24.000Z","updated_at":"2025-11-15T15:20:56.000Z","published_at":"2020-08-01T16:58:23.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46054\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46057\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46060\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e involve the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#, or the product of prime numbers up to and including the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In particular, the second of these deals with the last non-zero digit (LNZD) in the primorial. For example, the first eleven primorials are\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, and 6469693230\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, {\\\\rm and}\\\\,\\\\,200560490130 \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAfter the second primorial, the LNZD is 1, 3, 7, or 9. This observation holds for larger \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as well because prime numbers greater than 5 end in 1, 3, 7, or 9 and products of numbers whose LNZDs are 1, 3, 7, or 9 also have a LNZD of 1, 3, 7, or 9. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMathematicians \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehave determined\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e that the last digits of prime numbers repeat themselves less often than expected. If these digits occurred randomly, 25% of the primes that follow primes ending in a 1 should also end in a 1. However, as verified in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45994-investigate-the-frequency-of-last-digits-of-prime-numbers\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45994\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, for primes up to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"100 million\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e10^8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the actual frequency is less than 18%. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat about primorials? What happens when basic multiplication facts interact with the frequency of last digits of primes? Will the frequencies be similarly skewed? Or will they all approach 25%? \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes as input an integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and computes the frequency of the last digits of primorials between \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# = 30 and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#. Return a matrix whose rows correspond to the digits of the first primorial and columns correspond to the digits of the next primorial. Please remember to (a) omit \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# = 2 and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# = 6 and (b) account for the primorial following \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#. For example, if your function is given \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 10, then it should return\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    0.3333    0.3333         0    0.3333\\n    0.6667    0.3333         0         0\\n         0    1.0000         0         0\\n         0         0    1.0000         0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThat is, with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 10, we consider eight primorials (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e# through \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_10\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_{10}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#). Of these eight, three have a LNZD of 3 (30, 30030, and 6469693230). Two of those (i.e., 66.7%) are followed by primorials with a LNZD of 1, and one (33.3%) is followed by a primorial with a LNZD of 3.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46057,"title":"Find the last non-zero digit in a primorial","description":"\u003chttps://www.mathworks.com/matlabcentral/cody/groups/2001/problems/45251 Cody Problem 45251\u003e asked us to find the last non-zero digit in a factorial. For this problem consider the primorial p_n#, the product of primes up to and including the nth prime p_n. For example, for n = 12, the primorial is 7420738134810, whose last non-zero digit is 1.\r\n\r\nFind the last non-zero digit in the primorial p_n#. \r\n\r\nSee also \u003chttps://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial Cody Problem 46054\u003e \"Count trailing zeros in a primorial\".","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 123.75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 61.8833px; transform-origin: 407px 61.8833px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.75px; text-align: left; transform-origin: 384px 31.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/groups/2001/problems/45251\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 45251\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 284.733px 7.91667px; transform-origin: 284.733px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asked us to find the last non-zero digit in a factorial. For this problem consider the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.45px 7.91667px; transform-origin: 19.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#, the product of primes up to and including the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2333px 7.91667px; transform-origin: 27.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 56.7833px 7.91667px; transform-origin: 56.7833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example, for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.75px 7.91667px; transform-origin: 143.75px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 12, the primorial is 7420738134810, whose last non-zero digit is 1.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.6333px; text-align: left; transform-origin: 384px 10.6333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.267px 7.91667px; transform-origin: 132.267px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFind the last non-zero digit in the primorial \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p_n\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 7.91667px; transform-origin: 5.83333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e#.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2333px 7.91667px; transform-origin: 27.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problems 46054\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.783px 7.91667px; transform-origin: 124.783px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \"Count trailing zeros in a primorial\" and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e46060 \"Identify primorial primes\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = primorialLastNonzeroDigit(n)\r\n  y = f(n);\r\nend","test_suite":"%%\r\nn = 1;\r\ny_correct = 2;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 2;\r\ny_correct = 6;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 3;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 4;\r\ny_correct = 1;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 12;\r\ny_correct = 1;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 29;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 53;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 89;\r\ny_correct = 7;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 163;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 211;\r\ny_correct = 1;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 254;\r\ny_correct = 9;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 350;\r\ny_correct = 7;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 1379;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 37913;\r\ny_correct = 3;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))\r\n\r\n%%\r\nn = 1379137;\r\ny_correct = 9;\r\nassert(isequal(primorialLastNonzeroDigit(n),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":31,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-26T18:50:12.000Z","updated_at":"2025-11-15T15:07:03.000Z","published_at":"2020-07-26T19:43:27.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/groups/2001/problems/45251\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45251\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asked us to find the last non-zero digit in a factorial. For this problem consider the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#, the product of primes up to and including the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example, for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 12, the primorial is 7420738134810, whose last non-zero digit is 1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the last non-zero digit in the primorial \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e#.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems 46054\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \\\"Count trailing zeros in a primorial\\\" and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e46060 \\\"Identify primorial primes\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1688,"title":"Prime Sequences: AP-k Minimum Final Value","description":"Welcome to Fun with Primes. Today we will find the Minimum Final Value AP-k sequences for n_max=3:12 given the primorial and knowledge that the solution is of the form a + b * k# * n.\r\n\r\nThe AP-k of n sequence is n_max+1 primes of the form a + b * k# * n where n=0:n_max. The value of \"a\" is a prime and k# is the primorial.\r\n\r\nThe primorial k# is the product of all primes ≤ k, e.g. 10# = 2 · 3 · 5 · 7.\r\n\r\n*Input:* (k, n_max) \r\n\r\n*Output:* [a, b] for the equation Prime = a + b * k# * n,  n=0:n_max; Prime(n_max) must be the optimum minimum.\r\n\r\n*Value Range Limits:* [a\u003c150,000 , b\u003c8 ]\r\n\r\n*Example:* \r\n\r\n(13, 13) yields [31385539,14 ]; 31385539 + 14·13#·n (End Prime 36850999) \r\n\r\n*Commentary:*\r\n\r\n(13, 16) has a non-minimal end [17, 11387819007325752 ] to give Primes=17 + 11387819007325752·13#·n\r\n\r\nThe current June 2013 record for n is 25 via PrimeGrid:  43142746595714191 + 23681770·23#·n\r\n\r\n\r\n\r\n","description_html":"\u003cp\u003eWelcome to Fun with Primes. Today we will find the Minimum Final Value AP-k sequences for n_max=3:12 given the primorial and knowledge that the solution is of the form a + b * k# * n.\u003c/p\u003e\u003cp\u003eThe AP-k of n sequence is n_max+1 primes of the form a + b * k# * n where n=0:n_max. The value of \"a\" is a prime and k# is the primorial.\u003c/p\u003e\u003cp\u003eThe primorial k# is the product of all primes ≤ k, e.g. 10# = 2 · 3 · 5 · 7.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e (k, n_max)\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e [a, b] for the equation Prime = a + b * k# * n,  n=0:n_max; Prime(n_max) must be the optimum minimum.\u003c/p\u003e\u003cp\u003e\u003cb\u003eValue Range Limits:\u003c/b\u003e [a\u0026lt;150,000 , b\u0026lt;8 ]\u003c/p\u003e\u003cp\u003e\u003cb\u003eExample:\u003c/b\u003e\u003c/p\u003e\u003cp\u003e(13, 13) yields [31385539,14 ]; 31385539 + 14·13#·n (End Prime 36850999)\u003c/p\u003e\u003cp\u003e\u003cb\u003eCommentary:\u003c/b\u003e\u003c/p\u003e\u003cp\u003e(13, 16) has a non-minimal end [17, 11387819007325752 ] to give Primes=17 + 11387819007325752·13#·n\u003c/p\u003e\u003cp\u003eThe current June 2013 record for n is 25 via PrimeGrid:  43142746595714191 + 23681770·23#·n\u003c/p\u003e","function_template":"function [a,b]=APk_min_end(k,n)\r\n% a+b*k#*(0:n) are all primes\r\n a=0;\r\n b=0;\r\nend\r\n","test_suite":"tic\r\nn=3; p=3;\r\n[a,b]=APk_min_end(p,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(p))*n,23))\r\n%%\r\nn=4; p=3;\r\n[a,b]=APk_min_end(p,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(p))*n,29))\r\n%%\r\nn=5; p=5;\r\n[a,b]=APk_min_end(p,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(p))*n,157))\r\n%%\r\nn=6; k=5;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,907))\r\n%%\r\nn=7; k=7;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,1669))\r\n%%\r\nn=8; k=7;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,1879))\r\n%%\r\nn=9; k=7;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,2089))\r\n%%\r\nn=10; k=11;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,249037))\r\n%%\r\nn=11; k=11;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,262897))\r\n%%\r\nn=12; k=13;\r\n[a,b]=APk_min_end(k,n);\r\ntoc\r\nassert(isequal(a+b*prod(primes(k))*n,725663))","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":"2013-06-30T03:36:26.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-06-30T03:04:32.000Z","updated_at":"2013-06-30T04:28:44.000Z","published_at":"2013-06-30T03:36:26.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWelcome to Fun with Primes. Today we will find the Minimum Final Value AP-k sequences for n_max=3:12 given the primorial and knowledge that the solution is of the form a + b * k# * n.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe AP-k of n sequence is n_max+1 primes of the form a + b * k# * n where n=0:n_max. The value of \\\"a\\\" is a prime and k# is the primorial.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe primorial k# is the product of all primes ≤ k, e.g. 10# = 2 · 3 · 5 · 7.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (k, n_max)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e [a, b] for the equation Prime = a + b * k# * n, n=0:n_max; Prime(n_max) must be the optimum minimum.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eValue Range Limits:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e [a\u0026lt;150,000 , b\u0026lt;8 ]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(13, 13) yields [31385539,14 ]; 31385539 + 14·13#·n (End Prime 36850999)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCommentary:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(13, 16) has a non-minimal end [17, 11387819007325752 ] to give Primes=17 + 11387819007325752·13#·n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe current June 2013 record for n is 25 via PrimeGrid: 43142746595714191 + 23681770·23#·n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" 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