Problem 57323. Easy Sequences 84: Digits of Powers of 5

Created by Ramon Villamangca

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:hyperlink w:docLocation=\"https://oeis.org/A008566\"><w:r><w:t>OEIS sequence #A008566</w:t></w:r></w:hyperlink><w:r><w:t>, deals with the following series:</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:customXmlPr><w:r><w:t>A = \t1, 5, 2, 5, 1, 2, 5, 6, 2, 5, 3, 1, 2, 5, 1, 5, 6, 2, 5, 7, 8, 1, 2, 5, 3, 9, 0, 6, 2, 5...</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>This sequence is the digits of powers of 5 listed side by side, that is:</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=\"false\"/></w:customXmlPr><w:r><w:t>5^0,\\ 5^1,\\ 5^2,\\ 5^3,\\ 5^4,\\ 5^5...\\rightarrow1,\\ 5,\\ 25,\\ 125,\\ 625,\\ 3125...\\rightarrow1,5,2,5,1,2,5,6,2,5,3,1,2,5...=A</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:rPr><w:b/></w:rPr><w:t>Given a digit </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> and index </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, write a function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>C(d,n)</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, that outputs how many times </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> appears in the sequence segment </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>A(1:n)</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><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>For example, it is clear from the list above that, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>C(1,10)=2</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>C(7,20)=1</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>C(5,25)=8</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>---------------</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:rPr><w:b/></w:rPr><w:t>HINT: </w:t></w:r><w:r><w:t>OEIS did not provide a close formula for finding the elements of sequence #A008566. However, this sequence is not without a pattern. For example: except for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k=0</w:t></w:r></w:customXml><w:r><w:t>, the last digit of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>5^k</w:t></w:r></w:customXml><w:r><w:t> is always </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>5</w:t></w:r></w:customXml><w:r><w:t>, except for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k\\le1</w:t></w:r></w:customXml><w:r><w:t>, the second last digit of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>5^k</w:t></w:r></w:customXml><w:r><w:t> is always </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>2</w:t></w:r></w:customXml><w:r><w:t>, except for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k\\le2</w:t></w:r></w:customXml><w:r><w:t>, the third last digit of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>5^k</w:t></w:r></w:customXml><w:r><w:t> is always either </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>1</w:t></w:r></w:customXml><w:r><w:t> or </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>6</w:t></w:r></w:customXml><w:r><w:t>, etc... You may need this pattern as the use of big integer class is not required in this problem.</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: 20.440001px; 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: none; white-space: normal; "><div style="block-size: 315px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 157.5px; transform-origin: 407px 157.5px; vertical-align: baseline; "><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; "><a target='_blank' href = "https://oeis.org/A008566">OEIS sequence #A008566</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: 0px 0px; transform-origin: 0px 0px; ">, deals with the following series:</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: 0px 0px; transform-origin: 0px 0px; "> <img 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" width="450" height="18" style="width: 450px; height: 18px;"></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: 0px 0px; transform-origin: 0px 0px; ">This sequence is the digits of powers of 5 listed side by side, that is:</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: 0px 0px; transform-origin: 0px 0px; "> <img 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" width="584" height="19" style="width: 584px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; ">.</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: 0px 0px; transform-origin: 0px 0px; ">Given a digit d<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: 0px 0px; transform-origin: 0px 0px; "> and index 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: 0px 0px; transform-origin: 0px 0px; ">, write a function <img src="data:image/png;base64,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" width="48" height="19" style="width: 48px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; ">, that outputs how many times d<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: 0px 0px; transform-origin: 0px 0px; "> appears in the sequence segment <img src="data:image/png;base64,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" width="53" height="19" style="width: 53px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; ">.</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: 0px 0px; transform-origin: 0px 0px; ">For example, it is clear from the list above that, <img src="data:image/png;base64,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" width="81" height="19" style="width: 81px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; ">, <img src="data:image/png;base64,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" width="81" height="19" style="width: 81px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; "> and <img src="data:image/png;base64,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" width="81" height="19" style="width: 81px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; ">.</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: 0px 0px; transform-origin: 0px 0px; ">---------------</div><div style="block-size: 84px; 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 42px; text-align: left; transform-origin: 384px 42px; 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: 0px 0px; transform-origin: 0px 0px; ">HINT: <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: 0px 0px; transform-origin: 0px 0px; ">OEIS did not provide a close formula for finding the elements of sequence #A008566. However, this sequence is not without a pattern. For example: except for <img src="data:image/png;base64,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" width="36" height="18" style="width: 36px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">, the last digit of <img src="data:image/png;base64,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" width="15" height="19" style="width: 15px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; "> is always 5<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: 0px 0px; transform-origin: 0px 0px; ">, except for <img src="data:image/png;base64,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" width="36" height="18" style="width: 36px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">, the second last digit of <img src="data:image/png;base64,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" width="15" height="19" style="width: 15px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; "> is always 2<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: 0px 0px; transform-origin: 0px 0px; ">, except for <img src="data:image/png;base64,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" width="36" height="18" style="width: 36px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">, the third last digit of <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAmCAYAAADTGStiAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAHqADAAQAAAABAAAAJgAAAABbEMRCAAACLUlEQVRYCe2Wu0scURSHVzdiYUwjKBFE1EQRRAk+CmMrFtZWpjSdwcbGFPoHaGFhJWgINqniCzsTgosGRIQVtogaIokYQlIYQXzg6/u5M3gzzLozo0ya+cG3c1/nnL2Pc2disUjRCvhbgTjDX8EYLMMLCEU5RHkOKbiEeghVX4j2C/RHXPXAtfXfxiaqbg4KaR+0WDJMHlOuhnegWQdSB1YyzsQefXkOz9pXje9xtPuqfrCcZAo85OLtrWVTwbMdVmAVFqEBruW2hHZfI4U1+AQJcOoPDRNw6OjYpX4CVfAQkqAJ9MMBZJX26ByeZh15M6CWolZnHErhM7wEz6pk5Bloxo88W6VzWIGH4QfojPiSLgB7XzXrDRiBbHk5Z9jJvgU8q4iR2jc7sPM5S1+JizelpvZwC7pAdlPgWeWMnIdvcAHOwKp/B+2hqVYq6huFXNiGUygDHWLdaJ5VwMhmeANacvNPrFOPg60hCurvtBp0qFRfgNfQC4H0DCvlpBm82/D0kbJmqBSSNOsEaPwMmH+Sqj/phG+CHVyXwm1SMD/peJuvWB29SjUF10EKVSmiKfDfoFG1D0GkgyXtpx/+f4MGPrZC/fQfMm0RNLD9ltFlEpraiKT9VeoU31fUfBwpRfSCn4YnYErXaRIUuM/suGtZLwE5tTmiPAA1oPvXzmG9W+9dk3h0u6PV9h70gXBnZfoC0WeLLgp9uOmNswNf4TdEilYgWoFoBf7vClwB1wWQ1I5VqEsAAAAASUVORK5CYII=" width="15" height="19" style="width: 15px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; "> is always either 1<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: 0px 0px; transform-origin: 0px 0px; "> or 6<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: 0px 0px; transform-origin: 0px 0px; ">, etc... You may need this pattern as the use of big integer class is not required in this problem.</div></div></div>

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Last Solution submitted on Jan 16, 2023

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Problem 57323. Easy Sequences 84: Digits of Powers of 5

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Problem 57323. Easy Sequences 84: Digits of Powers of 5

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