Problem 46087. Investigate the frequency of last non-zero digits of primorials

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Cody Problems </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial\"><w:r><w:t>46054</w:t></w:r></w:hyperlink><w:r><w:t>, </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial\"><w:r><w:t>46057</w:t></w:r></w:hyperlink><w:r><w:t>, and </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes\"><w:r><w:t>46060</w:t></w:r></w:hyperlink><w:r><w:t> involve the primorial </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"p_n\"/></w:customXmlPr><w:r><w:t>p_n</w:t></w:r></w:customXml><w:r><w:t>#, or the product of prime numbers up to and including the </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>th prime </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"p_n\"/></w:customXmlPr><w:r><w:t>p_n</w:t></w:r></w:customXml><w:r><w:t>. In particular, the second of these deals with the last non-zero digit (LNZD) in the primorial. For example, the first eleven primorials are</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, and 6469693230\"/></w:customXmlPr><w:r><w:t>2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, {\\rm and}\\,\\,200560490130 </w:t></w:r></w:customXml><w:r><w:t> </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>After the second primorial, the LNZD is 1, 3, 7, or 9. This observation holds for larger </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> 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. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Mathematicians </w:t></w:r><w:hyperlink w:docLocation=\"https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html\"><w:r><w:t>have determined</w:t></w:r></w:hyperlink><w:r><w:t> 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 </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/45994-investigate-the-frequency-of-last-digits-of-prime-numbers\"><w:r><w:t>Cody Problem 45994</w:t></w:r></w:hyperlink><w:r><w:t>, for primes up to </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"100 million\"/></w:customXmlPr><w:r><w:t>10^8</w:t></w:r></w:customXml><w:r><w:t>, the actual frequency is less than 18%. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>What 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%? </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function that takes as input an integer </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> and computes the frequency of the last digits of primorials between </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"p_3\"/></w:customXmlPr><w:r><w:t>p_3</w:t></w:r></w:customXml><w:r><w:t># = 30 and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"p_n\"/></w:customXmlPr><w:r><w:t>p_n</w:t></w:r></w:customXml><w:r><w:t>#. 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 </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"p_1\"/></w:customXmlPr><w:r><w:t>p_1</w:t></w:r></w:customXml><w:r><w:t># = 2 and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"p_2\"/></w:customXmlPr><w:r><w:t>p_2</w:t></w:r></w:customXml><w:r><w:t># = 6 and (b) account for the primorial following </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"p_n\"/></w:customXmlPr><w:r><w:t>p_n</w:t></w:r></w:customXml><w:r><w:t>#. For example, if your function is given </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> = 10, then it should return</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![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]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>That is, with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> = 10, we consider eight primorials (</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"p_3\"/></w:customXmlPr><w:r><w:t>p_3</w:t></w:r></w:customXml><w:r><w:t># through </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"p_10\"/></w:customXmlPr><w:r><w:t>p_{10}</w:t></w:r></w:customXml><w:r><w:t>#). 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.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div 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; "><div 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; "><div 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; "><span 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; ">Cody Problems <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/46054-count-trailing-zeros-in-a-primorial">46054</a><span 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; ">, <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/46057-find-the-last-non-zero-digit-in-a-primorial">46057</a><span 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; ">, and <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/46060-identify-primorial-primes">46060</a><span 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; "> involve the primorial <img src="data:image/png;base64,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" alt="p_n" style="width: 15.5px; height: 20px;" width="15.5" height="20"><span 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; ">#, or the product of prime numbers up to and including the n<span 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; ">th prime <img src="data:image/png;base64,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" alt="p_n" style="width: 15.5px; height: 20px;" width="15.5" height="20"><span 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; ">. In particular, the second of these deals with the last non-zero digit (LNZD) in the primorial. For example, the first eleven primorials are</div><div 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; "><span 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; "> <img 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"><span 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; "> </div><div 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; "><span 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; ">After the second primorial, the LNZD is 1, 3, 7, or 9. This observation holds for larger n<span 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; "> 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. </div><div 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; "><span 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; ">Mathematicians <a target='_blank' href = "https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html">have determined</a><span 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; "> 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 <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/45994-investigate-the-frequency-of-last-digits-of-prime-numbers">Cody Problem 45994</a><span 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; ">, for primes up to <img 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"><span 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; ">, the actual frequency is less than 18%. </div><div 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; "><span 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; ">What 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%? </div><div 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; "><span 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; ">Write a function that takes as input an integer n<span 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; "> and computes the frequency of the last digits of primorials between <img src="data:image/png;base64,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" alt="p_3" style="width: 15.5px; height: 20px;" width="15.5" height="20"><span 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; "># = 30 and <img src="data:image/png;base64,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" alt="p_n" style="width: 15.5px; height: 20px;" width="15.5" height="20"><span 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; ">#. 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 <img 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"><span 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; "># = 2 and <img src="data:image/png;base64,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" alt="p_2" style="width: 15.5px; height: 20px;" width="15.5" height="20"><span 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; "># = 6 and (b) account for the primorial following <img src="data:image/png;base64,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" alt="p_n" style="width: 15.5px; height: 20px;" width="15.5" height="20"><span 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; ">#. For example, if your function is given n<span 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; "> = 10, then it should return</div><div 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; "><div 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; "><span 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; "> 0.3333 0.3333 0 0.3333</div><div 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; "><span 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; "> 0.6667 0.3333 0 0</div><div 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; "><span 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; "> 0 1.0000 0 0</div><div 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; "><span 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; "> 0 0 1.0000 0</div></div><div 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; "><span 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; ">That is, with n<span 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; "> = 10, we consider eight primorials (<img src="data:image/png;base64,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" alt="p_3" style="width: 15.5px; height: 20px;" width="15.5" height="20"><span 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; "># through <img src="data:image/png;base64,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" alt="p_10" style="width: 20.5px; height: 20px;" width="20.5" height="20"><span 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; ">#). 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.</div></div></div>

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Problem 46087. Investigate the frequency of last non-zero digits of primorials

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