Problem 52978. Easy Sequences 44: Finding the Smallest Number whose Cube is divisible by a Factorial

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:r><w:rPr><w:b/></w:rPr><w:t>Given a integer </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>, our goal is to find the smallest integer </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</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, such 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>n!</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> divides </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^3</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, 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>n=6</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>k=60</w:t></w:r></w:customXml><w:r><w:t>, because </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!\\ | \\ 60^3</w:t></w:r></w:customXml><w:r><w:t>, (since </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>\\frac_{60^3}_{6!} = 300</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>60</w:t></w:r></w:customXml><w:r><w:t> is the smallest number that has this property. Please present the value 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>k</w:t></w:r></w:customXml><w:r><w:t> in modulo </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>59999999</w:t></w:r></w:customXml><w:r><w:t> (a prime number).</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: 20.440000534057617px; 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: normal; text-decoration: none; white-space: normal; "><div style="block-size: 83px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; 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; text-align: left; 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; ">Given a 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; ">, our goal is to find the smallest integer k<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; ">, such that <img src="data:image/png;base64,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" width="15.5" height="18" style="width: 15.5px; 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; "> divides <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; ">.</div><div style="block-size: 53px; 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; text-align: left; 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; ">For example, for <img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; 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; ">, <img src="data:image/png;base64,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" width="43.5" height="18" style="width: 43.5px; 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; ">, because <img src="data:image/png;base64,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" width="54.5" height="20" style="width: 54.5px; height: 20px;"><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; ">, (since <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHYAAABACAYAAADPhIOhAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAdqADAAQAAAABAAAAQAAAAADYl14/AAAILElEQVR4Ae2ba6xdRRWAS4uForaA2mAFcwWtolheGhQh1RBiggYQkoKoiRgTH9HwMGJAExOMD36hURoDIVUDVXyAJoIKFIsghioWBEERTW0JIGBtERCqqN93Outm7nSfc3vOvvf0nptZyXdnZs1jz15rz/O0c+ZUqRaoFqgWqBaoFqgWqBaoFqgWqBaoFpiVFthtSG+1H885C5bBY3ADXAlVRtgCi+n746BD74H/JT5PWGWELbCKvp8Nc9M7HE+oc3X0sGaM9OgaTJUFnIK/0NDYg+j+Bbs35FXViFpgIf3+N3x3RPtfu91gAUfoFfAb2Kchv6pGzAKupe+GX4Pr67NwPlSZJgvMm6Z2y2afh+IYeAgOhH3hOLgZNkCVXWCBI3nmhfBTuAYWQMheRM6Ab8BV8AHYA3qJa6ztOHK/3qtgzZseCyyh2WtBB1wGB0EuOvBG2AhvgeXgbtc686GXHEWm7V7Xq1DNm3oLLKVJHeZaeFKX5i9Br3NOzvJPSbovZrqmqFPzf6GO2CbrTJNuLu3eDjrt412e8Xz0T8CTkJ9FddhT4OXDntBNjiDD9t/ZrUDVT70FzqVJjb6uR9MrUpk1DWVuSnknEr4cfgEXw76gLIJbYZWJKsOzwGYepWPdDK2G28DNjs4M8ahimW+FIgvVmXcO7A+ObNP/ANdkjzynQZVptEA+jfqYMYiLg6OJ69g/w0fBtfQwuABeBsqW7cGEv1tTytHqZupFcAi42boXdHSVabZA6dhD0/MeIXwNuF4qP4DfwifhUoj18xnipWxLCtdhxevD9Z1Y/TM0C7hRymVJSujEcKqqO8Ep2Q/Bo82joCzcHkz4Gw79+wRtTQzVAuWI3ZSe7sVDKQ+g0KkvhYdT5t4pzIPQRZk8b5TjLiknwOvAD9r77l/BfTCZeKX6dngzOHjuBi987ofJpE3d8ba9hHCjoxNLuRqFeafDwSnuD+elxI/py8qMEU6fSd+dgXz/ki+j0/jdZDEZznjWc6b7PmxOaWfBGAhEd5A2dXdoLM6wry1yNpJ+DuIGSgd6yfASCLEjvoBf5GyRM3gR38kzuycFR9oWyB38WdJN4kb0LrCsJwLP+coecAuoXweLoJQ2dcu2Ounj+euN049hXkezfTdsJy5PaYPjYBt8zUSSlYRulpx2ZoO47DwN18OLsxeaT/xs0CbyO2iSm1GabxsxIKLcUiL+YwPznQ1LaVO3bGs8fTKxTeDXtgoegq9AfHFEO/Je/noc8iOQv4Jf+GyRC3kRR5QjrEl+j1LHyAuLAm/I8q4s8iK5OpVxJhwLJWGbulkz3aOvJMsFf0H3Ip0cp203FbNNLuKFyiUpf8e4ZXsc5W55BvFLIJz+oSIvkh/OynwplIRt6mbN1OggFnAUPwU6z5GXizNbbJDM906gSfxowvmPpgJt6jY9o+r6tMCplNcpj8B+Rd0DU575TrPlaI7i6s0P53qkalM32q3hgBY4gXpuiDwCHdDQxrHowlnekfeSfId9OAXb1B1/TnlBMZ4xAyPuHJdPQb9W0MaaAdtxZJ4FnwCnTPcebozcRHocDFkSEUId10vMj+OO9+t7ZoX7rbs+6o6SY911xk9/0f9BQh0yiHyMShdDaTNH2C/hYHgSlHxq3rpd1fXvP7OcxcT3ytL91h2vWnZyPGMGRtw1fm8K+nXHgG1cTr2fgOvg2+Az8AJQ9ofz4dMmEM+nId3W18jPfeC9QZu60eYOYawLww536MgIKMbo440Qtro/6/NJmX5Dpm+Kmh9tvIt4m7rj7c8dj9VIvxbYQAUd4U2b4m52fic2Z87fUmiwdxZviub5D1OgTd2m9qtuQAuspV6MOKdpxak5dN6ndxtA6s2Psm6e2tSl+uiJu2KPF23xjnsq5XM0pmO8kMjlbhLhsEPzjCy+LCvzQKZvU7fTTL5wZ+3OyOiu3hV3M0qMUp2Rixu9Q5JiOeFdeWaKuwkL8dgU0qZup43JdmzxoJkQOtIOmoKOXEMbj01BOzbh0ek+sF+ngG2HePy5NyV+SOh6XMqPUJyYlEsJ/5TibeqmJmrQywKnkalTPMosaCh4ATqn27UNeaq+CuY/B0dBLkeTUG++R6lS2tQt2+orvZDS74Nvwk3wHsjlIyQuhfwmJc8fhfgmOqnhZSO8H/yN9nBYCep9d5eJJpmH8jqwnKP3VaA4Ov8I6q+FpiWxTV2aHEw+SDWnM1/8VCh/r3SK3wp2fDmMqpxLx32HkifQ3QorYLLlTKd/B2J0biBue6a/DflNE8kJMnDdyTo14SkpcRHhefAzOB22QJNsQ+ka5NridDaq8mo6fhjsA37I94CjV+f0Ix5l3B0vAc+rd4Lt7Iy0qbsz7Xf+r40v9BdYNEmNePk3TVKuZu9iC7im+sXq2HfsRF9uS2X9QqsM2QJz+3jemZT1VuQZ8N/kXA+uM17Oqy/lQRROx047VYZsgX4c605QcZfrurMW3ER8CtbBGOSiY6XftShvo8aHYIH1yUnuFEPcfF0NOu+yUKbwHMKfF7qaHJIF+hmxsVbekvVNh65M6bdmeqOOVjdQVXaBBfpxrBsnpTx3xeW1B/dcqmNzaww5vnsfz/NO9Eg4oKgTty46Mpc/kIjfKnN9jQ/BAv2M2NWpP3FpHd17Y4rcHooUPk0Yo7zIqsmZZIF5dOYGeBaOzTrmLxoegeIeNLK8ofG484pQ1HDmWsD1VUduhqtgDbjGeq4t5Q4U/4HXlxk1PXMtoIOPgbEeXfSmanGP/JpVLVAtUC1QLVAtUC1QLVAtUC1QLVAtMI0W+D+OciNNGtOQ9QAAAABJRU5ErkJggg==" width="59" height="32" style="width: 59px; height: 32px;"><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; ">), and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAJKADAAQAAAABAAAAJAAAAAAqDuP8AAADE0lEQVRYCe2WS4hPURzHr+dkyiuNGJLIhAhl8khJFCt2pigL5blhoWRhYWdNEqVQHiGyIAsLUtKIvENhwpDXLLyf8fmO+5v5zfmfMzcjZXG/9ZnzO7/H+d97XneyrFQ5A+UM/NsZ6NaF4SdTMw/GQR2cg80QSmPPhxlQCzfhDNyHv1YfRtgIzfAT3sBeWASKhRqM4xoo9yUcg5a8f5F2AHRZs6h8Ahpc7IBqSGkggeug3LPQC6QquADyN0J/+GMtpuIzaJAXsACKdJ4E5X+E0UGylvgTKH48iBV2p5PxA1T8DbQXijSVBOWLA4nkg3lcY49M5FS4e+O5BTb4hoqMuEPLaTWr4inZapezNZFT4d7iii5h68QUSXvFNq4eamyiYDx+e2ht+EKp4CtY0dq8YgjtRFAb0yicVqPlSL2E/LYVlD8I2tS9zWo3lmHayZBXG1Nv/hxu5O0V2pXgNcx13mLrx2KS/50LjHB21LQja2/7nqwjsB90uZlfrfaMqQHDYo/MmWibXK7usqQ0Y9/BBr6LXRNkrwlyFubxdbRWp0uxM/kXW+ETwyXTevZwCeuxX7m+zJ2wz/k25bbuF1Nq/1i8pxm0/vey8IHC2/OyK/TmLtepx9YP6OI0heOY31r/ufF1FQ+kjeulCzGmqzi1PJLeUBvTD1z0rfLxDr8ZztAHBn4NJn0kY9I+83n96D91ieqHY1tYfsVNz8xQGyt64BJSd45S/D55SF8PpNtdUkx3VkwTcFqtfuuxT4o90AmXMNfZ3tSA9pZaKt070tHfTevf2c725hzXSX3vXEqWDaVnR/92h0h7ZxKmHfHd7e7Wf9rM71/MpWQnXe0YH+jM1tG2gZdGErflcR11f0MrdXse0+dhmhxOM7Hts7HH+QtNnZzDoIf6Ag1QBVri5aBB9TBLIJRqT4Nq74DNQh32vdx/itbfRXSLpQIthx5Ig+sEtuS2Nu8USKkvAb2QzUYTtsZQ/xBUQ1S226PB3FlDWw/DoRkaIby9cUWl+0n7rRZ03+iT0uFU0S9VzkA5A+UM/Fcz8Avy8MG5/7ZTcwAAAABJRU5ErkJggg==" width="18" height="18" style="width: 18px; 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; "> is the smallest number that has this property. Please present the value of k<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; "> in modulo <img src="data:image/png;base64,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" width="63" height="18" style="width: 63px; 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; "> (a prime number).</div></div></div>

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Problem 52978. Easy Sequences 44: Finding the Smallest Number whose Cube is divisible by a Factorial

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Problem 52978. Easy Sequences 44: Finding the Smallest Number whose Cube is divisible by a Factorial

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