Problem 49788. Carmichael Number

Created by Gifari Zulkarnaen

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://en.wikipedia.org/wiki/Carmichael_number\"><w:r><w:t>Carmichael number</w:t></w:r></w:hyperlink><w:r><w:t> is a composite number </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:t> which satisfy following relation:</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>b^{n-1} \\equiv 1\\ (\\textnormal{mod}\\; n)</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>for all integers </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>b</w:t></w:r></w:customXml><w:r><w:t> which are coprime to </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: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, the smallest Carmichael number is 561. It has prime factors 3, 11, and 17, thus it is a composite number (not prime and not 1). The relation </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>b^{560} \\equiv 1\\ (\\textnormal{mod}\\; 561)</w:t></w:r></w:customXml><w:r><w:t> is true for all integers </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>b</w:t></w:r></w:customXml><w:r><w:t> that are not divisible by 3, 11, or 17 (coprime to 561).</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>Build a function isCarmichael(x) that returns true if x is a Carmichael number and false otherwise.</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>Hint: 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>b^{n-1}</w:t></w:r></w:customXml><w:r><w:t> can become a big number, using a </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Modular_exponentiation\"><w:r><w:t>modular exponentiation</w:t></w:r></w:hyperlink><w:r><w:t> algorithm may help.</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: 20.44px; 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 solid rgb(0, 0, 0); white-space: normal; "><div style="block-size: 211.4px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 105.7px; transform-origin: 407px 105.7px; vertical-align: baseline; "><div style="block-size: 20.8px; 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 10.4px; text-align: left; transform-origin: 384px 10.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><a target='_blank' href = "https://en.wikipedia.org/wiki/Carmichael_number"><span style=""><span style="">Carmichael number</span></span></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; "><span style=""> is a composite number </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><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; "><span style=""> which satisfy following relation:</span></span></div><div style="block-size: 20.8px; 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 10.4px; text-align: left; transform-origin: 384px 10.4px; 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; "><span style="">    </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAN8AAAAlCAYAAADC8DxTAAAKU0lEQVR4Xu2cZaw1SRGGnwWCu1uCu7st7u7u7i6LyxJcd/Hg7u7uwSFY0ECwIMEdguWBKtLf3LEzd849cruT++P75nRPT3W9JW/VzEHUUSVQJbASCRy0krvWm1YJVAlQwVeVoEpgRRKo4FuR4OttqwQq+KoOVAmsSAKbDj73f07gEOBZwCdXJMdNvu3RgIsDHwP+uskPssK9HxW4BPBx4C9j97HJ4Ds1cDfg2sBpgYMr+MYe+/9/dyLgocBzgW8sPLtOKCVwduDmwBOAX48RzSaDT2vzt1CeQyv4xhz3Ab85CfBI4PAKvIVl1zXhLMCtxgJwLPj83bHDpf59tq3Os9DDgAq+xWRpqKmF/jTwauDfi01f618fCRAEZwTeDuy1vt4oIrEnAv/ok1Qf+K4KXAU4B3Bu4I/A1YEvrJnotwF8RwAuCtwBeAzwrSXLWAW5FHBf4E9LvtdeLH9E4JLA9UJnTwU8D7j3IjnYTBvVsD0VeA/w1qngc56h3ZOBu8ZixrS/7FjwWMCVgfNHAv9M4H3AvYA7Ae8G7g78ZqaHzGU2GXwJuvtE7vpl4IZLBt/JgOcDWmZJlm0YGZn5bM8GLr1C8CnPCwIPB24P/LRLwENh59GBw4DbAY8DVPR/9pxW/v4CAbprAC8L0F0EuD7w9ZlPe1PBdzbgZsBPIqK4ArAX4LtteAkN6h9mPotVL6fXeRpwxxWDT0ck+/5R4IVTwaf7flWERNcE3jYg3RMCLwf+DPw4knk95QsCtB74r2Y+oU0F35GLfOQWwEv3AHzHibP4QCjnzEex8uXWBXwKQgNwRUBj1xrtDXm+y0foONYiny+SXG+uy31XJL6vBfwz1EnPmaAZc2J98fumgq987psCr9gD8Hk+bwTM+SRbtm2sE/guDLwGuG4XT9IHPq89BHh0eLMxYYpot2Zkwmn9yIKjoaeKZTJsDphDRkq6e8z4LfDVjpC3gm+MBP/3G89HL3sT4Afjp23ML9cJfBk16jhMvXaMPvBliCJoJASMpftGPriV/lsCn4P/Nm4/ArgScGPg+0s4xt2A70zhkc+1y32ZXGukpo698Hwygu7xDJHD/65ns9L1pwRMNc4TrKHkkGd4NcCc/kvBB7wT+Bdwgrh+rWDIvxjkR15v3s71ZNHVFZskTFmOB7w3PMZnYt0uvbWzSQLvzNGZ87Mw/BoXo65VsZ2538TPd7q4kj7w+XBvAE4aLObngctFV8nFoor/pgDlj4BE+ndDKL8PYZpw/iJAaFeKQp1zVPCNk2aSYYb9fRS8Z6uxlBzTOAkGmev7AZ8CPguYjtw5CBvB4/neH/gI8L0IawWqhE4z4nG3EhIPDlZSAKlb1hrVj8cGiGUtPdtmt4iGwf27vm2FHwyQCmAbBiyRORYFn1g4bjzzecPAaLDuGTmbe7sHcB1A3Tay6+NA0hnlGvIgB4w+8KU1VqDWnww7Twy8MvI4NyXg3hzhjEVNywkPCgvkIWcOaBjqv6W256wTpmfVu25ye9leeD49k2enRxpirVUSFdxzczwnUpAkDvRQGlVb+zS81rMEU7KnNmQ8I0LcJgiOEbm/c2V7P9TQSY29BJ0gEoAPKGqRnvdtAqDqZLOOZipj04CRzKLg0yBY+3Rft449WQd9OiATrczkPqx9q/c6JqsAXRGEZbqnAKcDPN8dRGMX+GTiHh8HIGWqUPVoMnJW7Z1n7U7hONycvW1aQ91+JvOGOLKfznlg9F7O1U2hAC4Tlklhvwh4cYB7dHPrOKex9F/tJfhsPh8TIueezLVvAHyzIYUE59B1PWepfIay6oT/36W8eheJIfXO2nKCzPLM62MvbSxiWRpbFHz5eHr+T8S9rVvradVrgagXzlKGzzDEgwhY11sIfCcPK6klEO2WG7SCZbtMbtJNa8G0qnVMk8A6g6+L6R7ac9v1o4TXM3zrqxufJjzYhQoP5hsXAl5v8qj4axryOQiXJA3tUHGPOhOJx59HGiXo9MqCUSD2OZNJ4LtshJO6Yi2NG2rWKiRWLCKm53v/NL1b6az9RLgsGnZOAVd5mG3zfYtCQy530EdSZZFar2dNUnZW8BmF+X9dxn634Cvn2wVk+GyHlzmpQ3LndZGb6hX7XmHLtSSu3POOWl9b2FmWGHT7xsAmtc2RFmJsDXClKOu4+X4C31jCJUW1DPCV8u4DX5nLp36pi0ZXRmPLAl/ZVGJbmPmm90zvZtnMEFiDoEfsbB0DEnzKs5XgagNfWWJ4SbCbzebb0jLJMLlJX++pY5oEhhR92qoHzlqk1ODMoT1NuV6Cz5BNHqDrrYNksRN87slGDfN7Gda22tluPV82lXgvPZ+gSd1P+Ukojmm1zEhD/sMweUd42ga+LDFIllidt5zQHBmWmoBKJadbnkNJ9uMaQ4o8l0y8z11GFtmH9jTleiqkbVdDbGGCzzqhYZshYLY6LiPnKyM+PZr1SssqObJ10jLMmFbL00e98klhNHacYRv4UqjW49oK49LMMknWgbRcJsC97y3NpTlbvM6QIs/16BpWW54sQg99cmNoT1Oul97D4rOG+ystD1d6sPQyx4wShHMsb7WxnbvxfGXE18aUZtnMmuaYN0/0or6U0PkyQRN8ZYmhjUqVdpXlsQzRrMHMpSD7cZ0hRZ5LJtbYLJhbHhhi6ob2NPW6RJ0GwNd/uhjDZNsNU+1DtT4scE1vLMJ3Fe93A76hiC85jsSFdTwBacukHT7lSC9qjc9PnbS+M9kEX1liaLp2f6sntHdT1imFMJdirGKd/US4pHytoUlYdHbbxw+ngivv0zW/NOBGVxa0m9+PybnNyCrrxpYgnGud8GuF4vjit+AwL+zyjl161hfxlaC21dImAPdmz3Lbt280LPIlOqjOF2qb4LPX7i1RwS/Bp8W00Oj3KXzT2vLDNoSa6wA+lVGZatXbco25jZJpgx0rdoJ0KYZ6kTW1rvAwPUHbnofmq8x6PZ9Zr2bdz7Y0+z2tHxuuyTWY0pQNE65rzuX+ZSZtDvc5bHuzu8UuJ3XVl2kdeng7bWwO6fucRBnxtYWcZa5q5GAILLiy1NY8Iw2ceaGtc7aitY4m+Py37tcWHmsaMk0ymwrLoqPfxNiGzw7MrdBT1jOBl9CyUdm/HDakS0ZkH+UyWGQ/uaDny77Fcv92KllE1iNpnBx2o6jA7wCs1ZnHlNdtQTSUtCvF/kiv2QF1/Jb52YJW6pr5ke99CjQ9mZ7FjpquArZ9lvlS8Clijn2WGhM9k2/BuNdvj3QSZcTXVsbI/kw/OCUJY5nE82nbny2YAlQD0ptXD73PN0Wp6pz1l4DKpJI6su92/Xe9/jvMkNqGAEtwva2UFXzrf6DL2qHRjN/XMXf68JCiLGsTW7RuhsRnDc83GLFU8G3R6U94FAFoXiLdb2g5V9P7hK1s9BRxZO3SvFOSZRB4Pm0F30af+Sybl+Qwh/rhyPxolptu2SI2jCtDyalm2aHzUSv4tkwL6uNsjgQq+DbnrOpOt0wCFXxbdqD1cTZHAhV8m3NWdadbJoH/ACnP/UQmmrooAAAAAElFTkSuQmCC" width="111.5" height="18.5" style="width: 111.5px; height: 18.5px;"></span></div><div style="block-size: 20.8px; 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 10.4px; text-align: left; transform-origin: 384px 10.4px; 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; "><span style="">for all integers </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">b</span><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; "><span style=""> which are coprime to </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><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; "><span style="">.</span></span></div><div style="block-size: 62.4px; 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 31.2px; text-align: left; transform-origin: 384px 31.2px; 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; "><span style="">For example, the smallest Carmichael number is 561. It has prime factors 3, 11, and 17, thus it is a composite number (not prime and not 1). The relation </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="123.5" height="18.5" style="width: 123.5px; height: 18.5px;"></span><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; "><span style=""> is true for all integers </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">b</span><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; "><span style=""> that are not divisible by 3, 11, or 17 (coprime to 561).</span></span></div><div style="block-size: 20.8px; 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 10.4px; text-align: left; transform-origin: 384px 10.4px; 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; "><span style="">Build a function isCarmichael(x) that returns true if x is a Carmichael number and false otherwise.</span></span></div><div style="block-size: 20.8px; 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 10.4px; text-align: left; transform-origin: 384px 10.4px; 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; "><span style="">Hint: Since </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="28.5" height="18.5" style="width: 28.5px; height: 18.5px;"></span><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; "><span style=""> can become a big number, using a </span></span><a target='_blank' href = "https://en.wikipedia.org/wiki/Modular_exponentiation"><span style=""><span style="">modular exponentiation</span></span></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; "><span style=""> algorithm may help.</span></span></div></div></div>

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Problem 49788. Carmichael Number

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Problem 49788. Carmichael Number

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