Problem 57565. Easy Sequences 92: Number of Roots of a System of Linear Diophantine Equations

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:t>One of the tasks that Matlab is very good at, is in solving systems of linear equations. </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>In this problem we shall tackle a system of linear </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Diophantine_equation\"><w:r><w:t>Diophantine Equations</w:t></w:r></w:hyperlink><w:r><w:t> in which the roots are limited to certain range.</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 the number of variables </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:r><w:rPr><w:b/></w:rPr><w:t>positive 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>a</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><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>b</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> and a root limit, </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>L</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, create the function, </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>numRoots(n,a,b,L)</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>, that outputs the number of posible integer root sets of the following system of equations:</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>x_1+x_2+x_3+x_4...+x_{n-2}+x_{n-1}+x_n=a</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> </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>x_1-x_2+x_3-x_4...+(-1)^{n-1}x_{n-2}+(-1)^{n}x_{n-1}+(-1)^{n+1}x_n=b</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:rPr><w:b/></w:rPr><w:t> with:</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>x_1,x_2,x_3...x_{n-1},x_n \\ge L</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, if </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,a,b,L) = (2,10,4,1)</w:t></w:r></w:customXml><w:r><w:t>, the system of equations: </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>x_1+x_2=10</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>x_1-x_2 = 4</w:t></w:r></w:customXml><w:r><w:t>, where </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>x_1,x_2\\ge 1</w:t></w:r></w:customXml><w:r><w:t>, has only one root set, namely: </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>(x_1,x_2)=(7,3)</w:t></w:r></w:customXml><w:r><w:t>. Therefore, </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>numRoots(2,10,4,1)</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>=</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>1</w:t></w:r><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>If </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,a,b,L) = (4,20,6,3)</w:t></w:r></w:customXml><w:r><w:t>, two of the possible roots of:</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>x_1+x_2+x_3+x_4=20</w:t></w:r></w:customXml><w:r><w:t>; and</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>x_1-x_2+x_3-x_4=6</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> where: </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>x_1,x_2,x_3,x_4 \\ge 3</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>are </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>(x_1,x_2,x_3,x_4)=(5,4,8,3)</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>(x_1,x_2,x_3,x_4)=(10,3,3,4)</w:t></w:r></w:customXml><w:r><w:t>. In fact, there are </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>16</w:t></w:r></w:customXml><w:r><w:t> possible root sets. Therefore, </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>numRoots(4,20,6,3)</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>=</w:t></w:r><w:r><w:t> 1</w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>6</w:t></w:r><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>There are no possible roots 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,a,b,L) = (10,55,10,-10)</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>(n,a,b,L) = (42,10,2,100)</w:t></w:r></w:customXml><w:r><w:t>, therefore in these cases the function should return: </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>numRoots = 0</w:t></w:r><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>NOTE: </w:t></w:r><w:r><w:t>As an added challenge, </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>only those solutions with </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/content/cody/about.html\"><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>Cody program size</w:t></w:r></w:hyperlink><w:r><w:rPr><w:b/></w:rPr><w:t> of less than or equal to </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>200</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> will be accepted.</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: 607.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 303.75px; transform-origin: 407px 303.75px; 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; "><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: 270px 8px; transform-origin: 270px 8px; unicode-bidi: normal; ">One of the tasks that Matlab is very good at, is in solving systems of linear equations. </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: 153px 8px; transform-origin: 153px 8px; unicode-bidi: normal; ">In this problem we shall tackle a system of linear <a target='_blank' href = "https://en.wikipedia.org/wiki/Diophantine_equation">Diophantine Equations</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: 145.5px 8px; transform-origin: 145.5px 8px; unicode-bidi: normal; "> in which the roots are limited to certain range.</div><div 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; "><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: 109px 8px; transform-origin: 109px 8px; unicode-bidi: normal; ">Given the number of variables 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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <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: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; ">positive integers, 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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, b<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: 58px 8px; transform-origin: 58px 8px; unicode-bidi: normal; "> and a root limit, L<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: 75px 8px; transform-origin: 75px 8px; unicode-bidi: normal; ">, create the function, <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: 68px 8px; transform-origin: 68px 8px; unicode-bidi: normal; ">numRoots(n,a,b,L)<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: 281px 8px; transform-origin: 281px 8px; unicode-bidi: normal; ">, that outputs the number of posible integer root sets of the following system of equations:</div><div 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: 384px 11px; text-align: left; transform-origin: 384px 11px; 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: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 249.5px; height: 20px;" width="249.5" height="20"></div><div 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: 384px 11px; text-align: left; transform-origin: 384px 11px; 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: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAvUAAAAqCAYAAAAztaReAAATTUlEQVR4Xu2d6+tuRRXH9Q8ou72yCOkCSUnRxSBKUKi0JIrKbiCCQVeEsNLKiMjupSBdLAoiym72JtJuUNCNtIyiwhdWhISvuvcH1Pp4nnWcs8/M3jN7zzx79vN8Nyz0/J6Z2TPfWTPznTVr1j7zDD1CQAgIASEgBISAEBACQkAIbBqBMzdde1VeCAgBISAEhIAQEAJCQAgIgTNE6qUEQkAICAEhIASEgBAQAkJg4wiI1G+8A1V9ISAEhIAQEAJCQAgIASEgUi8dEAJCQAgIASEgBISAEBACG0dApH7jHajqCwEh0C0CL7Ca3WRylcnt3dZSFRMCQkAIrIeA5smK2IvUVwRTRQmBA0PgddaeW03+3mm7Xmn1+kGH9Xu21ekSk3fucHuhSP2qGtS7HlO/z6yKkF6+BAHp1zz0NE/Ow200l0h9A1BVpBA4AAQ+ZW34sclXO27Lk61uHza5xuS3HdWTelGf75o830Skfr3O2YIeY6nkNOc1HW5Q1+u5bbxZ+nVqPzH38eTMx5onG+i4SH0DUFWkENg4AixUTMprWw990h+DkzTfNHlp5kKyz64Rqd8n2qe/qwc9frhV6yyTP09AwanT1SYXi9ivqzQFb5d+nQ4Wcx46/4wCHDVPjoPFGvdIkyeZfGQKV5H6KYT0uxA4LgTebs29aEcu1mo5BOd6k1+Z8P9TD8e4N3ZIiLRYTfVcu9/X1mOIzZUm15q8wyRng0ydzzF5YztYVHIlBKRfcSBF6ispmBXDpvF5Jo/dFfm9nHVZpL5eB6gkIbB1BCDHP9lNIlOWxRZt9QtTPol9zV6SQ+qpC25CfzJ5V6JilHOByWNM3JLEwgzp4nm6Se02i9S30JLpMtfWY9erh+6q+nr7bw6pRy/vMBm7WE3ZT92N0e/bf28weZ/JG0zYBJdYSKeRVIoYAtKvtF6I1NcdM65rlIqbqSz1dfFVaULgoBFgQobYrmEpfL+9994dujfv/ltC6iHrkPrnmPw00UsQIvfBxyXidybnmXCh9VUmte8PiNSvM1zW0mN0EOs8esUGEqLNk0vqScvmEwvd401SF9RDvbrU0t1iwkkVG1M2xLU3p+v0Yr9vlX6l+0akvq7eYui6bVfk2Np28q2y1NftgKWlOekoITNL36n820LAF/QSopDTwrWtT2Ed/7f7R+k4AJt/7YhRrM2Q9leYUO6PTLCeQqAgX1kTZg6QQRqR+vHFn0vEh6rHoYWtpI1Y6+8xGXPZ+Yf9jvzRBDc1NrH8Pw/H9SL1D1xSL8E+Z3j3Mk/2ql8i9TlalJ/G16d/WpaH5WQTqc9BaX9pROr3h/VW39SK1EN4sfL1cHw/l9Rj5fzKrh0xYgMRwiUi3Cw4GXqm/d0toz4Oc3XkEUHeMI9IfRrBQ9fjuaQLxNyX9nER+Lg095vd352w+ilV6SY4V7+3mE76le61Wvrl83SufmiezEXqgXSsT6zL2WNbpL4c5JY5ROpbonsYZbdYrLAO/s3kAyYpn/R9ojeX1Du5iVnnnGRh8bjQhOg+KTLE388uaPDdIvUFaJ1Ieuh6vITU++b0KTs9DcH1NQL/eY+U4+lrW6WLO7WjDNKvdGfU0i90fPjgBsbzlshvmifLBkhsAz9ZQg6pp+P+G5lcfOFLddTkyztKAHgPMhn64kJ2zjW5z2QfR5otSf1Yf6X6uKMuyqrKMbSxxWLlfnu9xFOfS+pREqzxXCAcXrD1sfVp+83vDLQmQyWW+mOYZ8NBfOh6vITU+2IeI+mOW3gHxI/oY5sAx1z6lbWEjCbqaZ7sTb8cuNbuN8ekx3zUzO+XhXdlnK/GePkZKVJPYS8xweeRB+tWGB2CS23+tcRerHulQ5YBerkJPogepSAkNX5hid8IJbSPD4PUJvV0Pn0ZhkUK+8tjfKMwwz4uxXOt9MfQxtZkyPWuhV/5HL1YQupZVHBbGLouOBkKiY+TIfT/fJN9X5Q9hnk21f8tSH1PeryEdIEZYyDcgDqO/J0L4aG7WOhCxt9v3yWWftW9syH9OqF3rl+xsd2C1K+px7HTiNI1LUq+MwrxO2CMd9azYXQ4ioCbYqQ6aXROkXr/6ItbssjsVgMIPSTxlyZcMNvqkZ+30Y/gaaP7LQHel3aTKm2MWf4y+qQ4SW1ST9v+bYKvsPtm+YJA9A+OcFk4uDzIJYysixjFrWqb4RjaGCLYggz55JFzcte2N0+UvpTUY4wYtiVGhvzC4XWW/tEmk+HCChvPHIkxJHUCcgzzbAqyQ9fjGqR+GJfaywz9a92qzzyOYeZnJk66pF91+UlP82QP+hUb23NIfc/zZOm9gRgm2f7wg8z+bgyxRNTCCAWB/6vJm3e8jSynlJ+ziHvBTBrfNrnJJHaBp3C96yq5LzCQ3NeaEBmjRdzqqUbXJvXh+8LLf37x4uOWoLZ1cqqNLX8/hja2IENeZs580LL/vOwlpN51IDyu9GPz4amiLya1TxshU5eZ+Gkm88p7TIhBngpTeAzzbKg7h67HS0lX7MTJT8jDTSJGDfSLhwU/NZ9Lv5bPXD3Nk73pl6NbQuq3ME/SnqXPD62AUoNRGMqS9QnDa3g6R53cUMtm/mQI3JxF3BUZCy+W3LH4uUsbX+OoIxWjeqxuIRn0BXjsiGluO91HP5X/1fYDJwNYaAhVlnrm+PiHkwBt5PSh1aXI0ouGsXb23sa5OjCVbwo7j0fNQP/OSGEld11KFqt9jNEapL4XV6Kp/g4XQ04Y9jHP5tZpSbqe9XhqHs5p99SReg3SFTtxyqlbLM0+1/G5dSzJJ/068aFAnjneEiVzfkm/4N2BF0DOx9ZKyj3UeTKFQejifgppDzJ4H/Knk5GFckh9SHjnKE9Jxy096siO5TmoVDgBzz0qyWln+J6c9Kk0c/vB8Z2LU26dS0MCxsrtvY25WJSmq4Ed7xy7NDesU8kEv48xeoykfp/zbKlOzknfsx7XmIen1oneSL30K67FLebJY9SvOXPEnDyHpscpDNwKz++pDyP6SfMpfC6H1IcKmhMdgyMVvtp4l0mpJXjpUcfYh2emFCj0X0rVGwsPn+T2i6eA+SGT3KMVxyZVF9ya/NLqnSMVxgVqzklC6GY0Fo+co5/3muCCxMMC9iaTlOvAsKrs1q+YAnzi99ZtDF/vn2ennaU6u7CZp2Wfwo4LnVzexqLrF+RidTjl8sxEJUtI/T7G6BJS7xaOVEzk2v1Vq7zSebbWe1PlMFfxASQPJIC+XVUw7/Ssx1PzcA62U0fqS0k9CzZPre9G9KZfjvFcviD9Wmapr61fOWOmRppe9bhG27yM8J4nXhWxOcDDUJOn2Kc+jJV5jRWQIrBU5GUm1+5IR20/1ZqgDctyUgehHl5O8rSkgdDgtoLbwxNNPrhraypPaZ1b+tRTF7/okzrOIY3fsObyYHgho1YbSzEpTZ/TxmGZTgS3oLMtfJE9CkzOJr+0P+akX0LqSzYoc+rWKk/uPNvq/WG5HhXro/bHP5hcYuL3A8bWgJK6HboeLyX1jIGac25P+oWetOYL0q/x0Vhbv0rG/pK0+9bjGu6mU656QzzCUJYpKz2bWj60yHOKq+nUIh6SXTKndg0+SIm0cqUJlvotECQH04mg/ztm5QNoImSEltxQwUqO8VJK3ZLUh4rC+1OnLkyGQyuvHwX1bv3MbeOQwHAxGuv3FnS2xWLVU6g2+mYpqWfeqmXhXLIA5eYtmWdzy1ySjvnwiybhaaCPrVque4euxzVIfa35qDf92gdfkH6NzwDMsbX0a8lcU5J3DT1e6m5K+6Zc9YYYhIbJVERC52SnbfynSD2FE/btLyYQdZ4pYucEYSsKw2L1NpMrTPziSYzwggXWquFHqKbCMZUobStS75sPTiI4RueJ9Q+D5lEmfG0zfDwSw/D2dUnbWqfNbeOwHvTfN3b6vQWdbbFYOQFJWQVa992w/CWkPhXfe99tKHnfnHm2pPzStOjY8Lsc4XHv1BqQ875D1+MlpL72eOxNv0L9aMUXpF/pUVhbv3LGe400a+jxUndT2j3lqjfEBs6NkTH2nQrS+piJelyMkXrILuEdsXjF/Jj427NMhu44rQZpDaUYluFE0BepoV89C9knTTjqSD0+efRqqffdrfvDxvzqcb/4mEnsq7nkv8dkLFxai74pKXNuG9HVc0xuMWFDl9ro4FbGKVQsXJyPg1sj+FGvsbwlbfS0LRYrymYiYRJZ+04BdZlL6n0897I5yenfufNsTtm108Ri/c99x6Hr8RJS76ciYVjWuTj3rl+t+IL0K60xNfVrrl6W5utdj0vbk0ofzhuxdSz8Pco5Q1LPgkgIrd+bPNgEoheGrwzj3BKuCJeFWHjLVoO0BmiQLNyD7jXBX/xbJi8y8TCYIeElLijHJsSXHruU6pcVa1ixa1nqGQA8vzD5nAmWaN98DWO5v9V+Y8cXI3N+MZiyIPU9PUvbiE8n/Us/n22SIvXub07bh/7E4YWW2K55LO9cLFstVtSVS5E9fINiLqlHt7nT0zLs7tx+83xL5lkmdPzbn7brJ/rrCSZcKocA5gQymFt/1/XSo+TU+w5dj5eQercOXjyjs5boF8arC0zoa3dh8zFFVVp8u6UVX5B+pZVniX7NUMlZWbamx7MaGckUhrIczufMKXBWngtNhh4V9/8QknofBP6e4S5g6Heesky3GqQ1QBuGWRuGTRz6ZE+FVfSFrpZlsAapH4bTGi7C4T0AME0t0u6WBFngSR0F1eiX0jJqtBF99wg7Xl7MUh/q/fD3EEtI/XDRG8tb2mZP32qxcgzWju8ejlFcxSCusROkGH5ssLnI3tsGNKzr0nnW+wlfyi+YcM8HIwWXplqOUQgfG7/kYlKo0Iesx37Ci8HA59jc6GFL15Sl+uXjDwPGWSZ8yfI8E1xPa61zoaq04gvSr/iAXKpfhcN8dvKt6fHshg4yh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style="width: 378.5px; height: 21px;" width="378.5" height="21"></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: 32.5px 8px; transform-origin: 32.5px 8px; unicode-bidi: normal; "> with:</div><div 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: 384px 11px; text-align: left; transform-origin: 384px 11px; 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: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 132.5px; height: 20px;" width="132.5" 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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</div><div style="block-size: 44px; 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 22px; text-align: left; transform-origin: 384px 22px; 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: 48.5px 8px; transform-origin: 48.5px 8px; unicode-bidi: normal; ">For example, if <img src="data:image/png;base64,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" style="width: 152.5px; height: 18.5px;" width="152.5" height="18.5"><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: 83px 8px; transform-origin: 83px 8px; unicode-bidi: normal; ">, the system of equations: <img src="data:image/png;base64,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" style="width: 77px; height: 20px;" width="77" 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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" style="width: 69.5px; height: 20px;" width="69.5" 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: 25px 8px; transform-origin: 25px 8px; unicode-bidi: normal; ">, where <img src="data:image/png;base64,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" style="width: 58.5px; height: 20px;" width="58.5" 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: 32.5px 8px; transform-origin: 32.5px 8px; unicode-bidi: normal; ">, has only one root set, namely: <img src="data:image/png;base64,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" style="width: 96px; height: 20px;" width="96" 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: 39.5px 8px; transform-origin: 39.5px 8px; unicode-bidi: normal; ">. Therefore, <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: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; ">numRoots(2,10,4,1)<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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> <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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">=<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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> <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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</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: 5.5px 8px; transform-origin: 5.5px 8px; unicode-bidi: normal; ">If <img src="data:image/png;base64,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" style="width: 152.5px; height: 18.5px;" width="152.5" height="18.5"><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: 90.5px 8px; transform-origin: 90.5px 8px; unicode-bidi: normal; ">, two of the possible roots of:</div><div 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: 384px 11px; text-align: left; transform-origin: 384px 11px; 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: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQwAAAAoCAYAAAALxsQHAAAJN0lEQVR4Xu2dR6s2NRTHfT+A3ZWIiGUhLnRhAVFBwYKCIAhWRFCw4speELGLBcSO7uxlI/aFgopYQUFwYUFEXFnxA+j5wRw5NzfJJPMkd555PQMH73uTSXL+Sf45JXPdtoM/joAj4AgUIrCtsJ5XcwQcAUdgBycMXwSOgCNQjIATRjFUXtERcAScMHwNOAKOQDECThjFUHlFR8ARcMLwNeAILB+Bg0WFHQc1PqxUZ3epf+Dwzjfy399y7zthVKLr1R2BNULgahnLtSK7BmO6Q/59/8jmP0rKbxSBMF4S2VnkEpFHRZ4S+SGmpxPGGs2+D8URqEDgM6l7aKb+21J2ToI0zpTfPycS1sFS+VLkj6HtTaThhFExQ17VEVgTBLAs7ha5WOTlgRT2lf+eLmItDiyNG4IxKynw60NEvgrKte3P5feHhfouiTDeksGfOID0+JpMXOth6GS9IA1zCmyPD6bwB8Mpttv2qGBnnXAhfhU5S+T5SF+KL0VYCiHGuo+ihCD1IZ7vh3YhpA17zQmj8+xWNu+EUQnY/7D6RaLzsSMHinVXrBVhrYuY9aFwfic/7CeyiVScMNZrxTlhrNd8rONoWCMfieSyIbqOGL8ljNvl39cPSqUsFIqxXM4Y6kEc/8UySggDE+dvkdDXwXTZU2Q0FdMI9Z4uiaalwknQlNMvFrRG+sSa6UkYuflKzXEPVXu6JOuiYw/catrUdYRLcoCIpkp1D4VEErZtCWcDsaQIA7PnNBFiBjxh1NQyVc60qVFyrG5rwjhZOjxP5AQRTUudIj+/MQyEGMIjQ1ku4jw27pry1oQBETKX6MhJwWPni/JXhrJkZLxGgYK6rQljTh3tHYYC1ZNVWh9IrFtSpGEs7Hez1mMBTx2gJYwN+ztFGEwCFoWmX2hIAyCQBQsQP4lBbQqMrIJc5t3WhKE62iCPAgyZPC1CThod3xmw6KTaf822Jgx0+0uEE0b9UgJaR4iQd8dHRUfMT4JjWxGEbE0Yc+poA4yrrI3We0jn+mgZlLWa/zGDLCWMDaRT4pJoJyys10QeFNl/FXQmvtuaMOwwbOT4Qil4T4Qcd/TyysTxl7zWmjBsn/bUwNpgIdwnEou0l4x1ap3WhDGnjhw6pDdXfdhTatmu2pbiG8u06V6OZU9sv5YIqwlDNxMnEyeQ9YlWVc6+r/5nqs0Hhk2MifRmpuMpMRW7mTh1b244gXaoYybs2VIZiwYX6LaMjlNMWLsI0BGrKczRt5pP+ko9B0nBYyIs2lMz9WJxs7HxbaWOY2OZq5z9yoGOFxAeeJYwcvsYC/v1QYFqwrCbqbXpZEG1/awCds7USrWbZNRVBhJ5d24TtvSEWVVta/pObWvqXZSt0nGqXj3fI151pwhp1zBJQb9b4pLYRW6DginF1Uz7QirUnGDES87PoHm4lBGcxNLBR0s9l0aYtWSSFMxcEBcL4VYRDSJySt4lck9JB1JnzITlZMBdoN1PM21ONWHHLu1ol5wwt4jo1WM272Ui2Q+TzHjpJ/WAobaLJZV63q3A1bZRqqN9hzF9IoKeNWu2cNq3pJrescjt0SlZkg1GQkkMw172uCYzieHV1NbZk54xDF0wbFYW8UmRKaYOY8CUxyXCtIbNIbHUO7UrpWcMg7Fofj1Mt9lxQhYQEh8m/SxyuQhB0VY69oxhlOoYzotm/WrX7JiLWTr/U1xM2zZ7lLjbuSK5WIhmT3i3NOi5gYDGCMNuJDpJXSelTKPVF8jPBIJqwR8Dtydh2IsqjGMPkfA0xdzbW8SeQGP38sd0Cst7EgbjJ3agT+okAufQStOoewyXWh17EkapjrHNBvHXrtm5XUz0KCUL6saynrH50722KdU+RhhsJHK3Pw4kkNpMtlNd9LXgjy28XoTBIrtK5HwRvnHgiW0msOCWXBhI0mu4Je7amI69CEOJDQtKvxOIzQ8HxF4iof+rgTTSsaVuSUrXXoRRqmM4LuaPz7unHHJjLubYfGv5VBeTQ5pDPGdZqBVEepWfvxWBHMl6cjDEHr2vsSmOlCMMNhIpRr5Yi8Ux+N2RIqH/viTC0EWmJ2cYxwDgh0VyH4IpkU0JtoaT1YMw1Eq8QjrDXI35+Jiq94rE0si6yFhcLVKwPQhjqo7gvY/IsyIcFq0PuVLCmFJPXWTILhdDY27fN3OXugWqY7AZkk1r2hIGm4ebnV+L7CRCRzb1Yu9j8AUbPlMsNbPOhAHIuEw/ieCfvypCak8vt9jNhN8Ow46lWDUA2+L0bUUYkD3PxyJPithFFd7HuFLKMT1jwT4N8tJW6jSqXeytCGNVHTmdmV/mmU8clkQYShZYjATdU89xUkCyINynahXHsp66B6LxSksYNoLKAEJ2Cf381Im6zoQRpm5DwEIfeCyNrLdEcx/y1GyoFoQR+tWhWWnjLowtlb5UV02vlOdM2BodWxBGCx1Z7+oKaHtLsDCULHJ/PMfOR8r1RH/a0LWrhwP3gJI4WMLAFMFs/VOE6H/ox7LQrhtGEivXQa4zYagOu5jFYsFV0CCCEr8SK4zLbDUpx9zmakEYVodUapJ+OH3GUpdqkeltxvCqcQ1RaN0WhLGqjvqJuM7bkghD564UeyyF2J0M3sfVPkaE9c7DeiAblqrf5X9k1IswUI4sxdinvaVArlqPRcbtU0zaVlfINS6Ey9QiXrCqjvq+Wl65tHppX5p+p37p/ZXStkvqabzD3oRcEmGU6NitzliWZErHvQhjylh6vcOi5z5G7Pptrz7nbpcY1hJM9jGcsAp5CHTqo9fVcc8eEln1XsTYGBZb7oRRP3WcUM+I8K1H7Z90r+9tfd6AMFrFaubUKozVxcayPRBjF4ydMOpg1YBTLHOCZZX1/+q6WqvamOxklOb4gncrgHCXpBDlHoQx9Zpt4ZBnrUY6iksvTwSj2Ef+jZuS+rPusw66onO9NUj85CYRLmkpSZKe3V7/+LITRuEiaUkYbJjjRfT7CnL75O75iKpVULBQrS7V7D38WAdjKdgug2rcKHOInvqX1kilMo8viiQj543HMEdzThiFqLckjMIuvZoj4AgsFQEnjKXOnI/bEZgBASeMGUD3Lh2BpSLghLHUmfNxOwIzIOCEMQPo3qUjsFQEnDCWOnM+bkdgBgScMGYA3bt0BJaKwL+OtXZHbYauAQAAAABJRU5ErkJggg==" style="width: 134px; height: 20px;" width="134" 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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; ">; and</div><div 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: 384px 11px; text-align: left; transform-origin: 384px 11px; 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: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 126.5px; height: 20px;" width="126.5" height="20"></div><div 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: 384px 11px; text-align: left; transform-origin: 384px 11px; 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: 57px 8px; transform-origin: 57px 8px; unicode-bidi: normal; "> where: <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAL0AAAAoCAYAAACrfPvAAAAHoUlEQVR4Xu2cR6stRRDH3/sAZlcuRAwLQVEMCKILBfNKVIzIBQUjLlSeeWXCtBExofAQzAguRAygCwNmFBQFA6LgyogfQP8/6YK6fbonnek55zo9ULzrmenqqup/VVfV9Lh9W72qBWZmge0z07eqWy2wrYK+gmB2Fqign92SV4Ur6CsGZmeBCvrZLXlVuIK+YuD/YIHDpMQuol9EP7QpVEHfZqF6f10tsJcEu110nmgPJ+T3+ntD9G5O8Ar6dV3SrSPX6RL1my4RdkSVAPxrogNEj4j+Eh0hOtfNcXwO+BX0I67ETFkBvlNEd4menAj8zEkac5voN2d30py3Q+R/Xf+emlqTCvqZInVEtS8Tr7sD0GBbGvzHaY6rRaQ1qeth/XiF6A/RnhX0I650ZbXJAqQbl4hunAD8benUDslwj+gT0dEV9BWppS2QAj8592OiL0pPHvhbpL88zLswbZf0hu3k74TQ++u3fURfi3xeVVI3jHqwKNWasrZVtmovINi6ydO0Jrl1LGCWbSnwk2PfUBj8zPut6CPRhTlc5kBPnnamiAKFi/zoKJH1QO/U3zeHe+Rwt5SwXOCJIteKTg4y8DMGNKW4/7ST9Qz9/eqM5MHZWS/sQzeDy68J918K9+J1LGim/1hPCX7r6NCyvEqUDcQ50GMotiOKhWeDZWy7APAY+GMRBUN2GxnJoiizmwiHs60L1ocHGZEDRXcX4aT2+0jTL7BZN3mI7rTsWOTvArixxzHBbuS2pBi08yjsksVdKWMFviXBD++TAjasX09QvDJgZkG1LunNP2EUhntF9KDowMJGyrFni34n3Dxf/x4qQlEUXMW1bvJYEYctiPrPix4QPbcK4yTmNPBTaNrld+2+YlonxwKeH5/d1bqA3vqwRA+ixEGiqXL4lBHMCYlgXMkKva/1lnh+neTxToh93hCVTD37mo3M4Y7gkIxFxjGdEv5kAxbxcfqF1mYX0PvoUTqV6WJEc0KeLZ3KbEV5zAmzfeouSo38TAx2wPiyqMQORLqHMwF8bLAQpLuA3kePLkUi9QDb16eFoox3wr01R27XQY6bRNQfGICd6hrR2EVuX3nsVTkLc6lo7FaeBYVsnzoBaGz1uajL+vbxhxTYHxKD0h0232hZCIxdQG8GQVlaTvdmtMbDzhbZC4pSXR06FY8GGXLnK6xjcZ+e+0p0msi6TU069FlQe7aLPNiGVONx0fuiY0WW13ZZgz5yET1xrGSUyzCiGUB3bizQrwrspp4P1L1BT+HxochaYU3Rw7oIvJljQUuA3jsgCubmYOGfEvmobuAcc9vvKg955jMiH+GsEzUW0LCHd0D+uwtvf4ygy/NNDrhqsMegT651W5QBPL+LfnSRqSmlYFLb7scGvTngrZpjQ0R7MueEbPHxywnG/xqs0qZD08LavT7yEHniLd3sNFZdYg5IgCKVawoKpgOBivza2s9DQR+DnbV/QTR26tZlXXjGWu1JDDaBnghAzkl3JJXX8xvbdJzutIHe0iB6y2+KWg/9B01xQBaTbkQqj0ben0VNOTtFnvWwrRZYlTx2UjBut5pdfwr26dIpMwe0miWV17Oz3B/Zm+d2ivYVsTsPAb2lU+ZkU520bHIA9KKtflbK8TzoiRREzy9Fu4owkq98fb+esxQc4Uy1L9tAby9QEDoGoFeEg0WHiOjjAmjAaUdF4/44464TNbUvGc98cRtrFfJYQEEfD2qT0eyQbLmFm/Dg+kD0hOhFkQWguF9/ve6x1fv2Jfa9WMTbS0tJh4AegNG0mArsVqTmPhZpTRs96H0rEGPG26736NR9W6g20JMu+S9deMmUal2Zk8E35RyeT5PzmFzWwz1BP/htd0p5ABrR2I53ILd/0RfXCMhOuhLvht7peSZ2jphPfN92Bjpb8LY1GwJ6HLXrbm1rscy//q08fAiKbwWG5wSscNQ5m1p50NuC/KkBqUHWAoR/E9M20Ns88GHxc6CHz4nBoPHHAow1PqmPCVJGJaLTPYnTsVXI41+ixN0k0xtg0lFJgd4+lQNwLHiqo2Z8UvcJMp+5ccuAfhkADxmL7nQJORtmFzuNpYOtDthWyA4Rqg30nifRPLWoQ+ZtGoNM+4lSzrMKeZjTonHuCx+i+U7R2Ec+4EszwBf6y4DeTrcus2ZTntQt8n846wp6clJSjdwXMMsY0Y+1SJ49ahoenkoeLxspJZErVYsQjambqJ/GvNjx7P2F8b1Af3B4kF2H9wh9QBinxUNkHauD1WnuVUV6gL4RRZtOAvd8yN4OZ0/cBX5TyROLn+vgkLfGhWdP1bOP+1op91Cf3J7C8sglhWtbnyXZbx6+CtBbl4hqv0s7bqjC9lbWijXPB1BZqjOVPLEe9sHDRbrh26y285U4l5Kz5TLpzdD1Wdm4EqC3ltLYL6f6GMkOHfEFjVX2Np7iuNS5oJyMpBSQ/3IIx+NqqzP66D302Qr6gZYDaBzmty/j2Z7ZtgBea0U9cM7UMPuChs5H7po0h5QQvs1G4YrTvSdqepE2oklaWVXQt5qoPlAtsIUtUCK92cLmqKLPwQIV9HNY5arjJgtU0FdAzM4CFfSzW/KqcAV9xcDsLFBBP7slrwr/C92UD0fNGf+rAAAAAElFTkSuQmCC" style="width: 94.5px; height: 20px;" width="94.5" height="20"></div><div style="block-size: 43.5px; 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 21.75px; text-align: left; transform-origin: 384px 21.75px; 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: 12.5px 8px; transform-origin: 12.5px 8px; unicode-bidi: normal; ">are <img src="data:image/png;base64,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" style="width: 159px; height: 20px;" width="159" 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: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; ">and <img src="data:image/png;base64,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" style="width: 166.5px; height: 20px;" width="166.5" 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: 58px 8px; transform-origin: 58px 8px; unicode-bidi: normal; ">. In fact, there are <img src="data:image/png;base64,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" style="width: 18px; height: 18px;" width="18" 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: 97px 8px; transform-origin: 97px 8px; unicode-bidi: normal; "> possible root sets. Therefore, <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: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; ">numRoots(4,20,6,3)<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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> <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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">=<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: 6px 8px; transform-origin: 6px 8px; unicode-bidi: normal; "> 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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</div><div style="block-size: 42.5px; 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 21.25px; text-align: left; transform-origin: 384px 21.25px; 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: 100px 8px; transform-origin: 100px 8px; unicode-bidi: normal; ">There are no possible roots for <img src="data:image/png;base64,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" style="width: 185px; height: 18.5px;" width="185" height="18.5"><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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" style="width: 175px; height: 18.5px;" width="175" height="18.5"><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: 82.5px 8px; transform-origin: 82.5px 8px; unicode-bidi: normal; ">, therefore in these cases the function should return: <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: 48px 8px; transform-origin: 48px 8px; unicode-bidi: normal; ">numRoots = 0<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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</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: 62.5px 8px; transform-origin: 62.5px 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: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; ">NOTE: <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: 76.5px 8px; transform-origin: 76.5px 8px; unicode-bidi: normal; ">As an added challenge, <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: 89.5px 8px; transform-origin: 89.5px 8px; unicode-bidi: normal; ">only those solutions with <a target='_blank' href = "https://www.mathworks.com/matlabcentral/content/cody/about.html">Cody program size</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: 85px 8px; transform-origin: 85px 8px; unicode-bidi: normal; "> of less than or equal to <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: 12px 8px; transform-origin: 12px 8px; unicode-bidi: normal; ">200<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: 26.5px 8px; transform-origin: 26.5px 8px; unicode-bidi: normal; "> will be accepted.</div></div></div>

One of the tasks that Matlab is very good at, is in solving systems of linear equations. 
In this problem we shall tackle a system of linear Diophantine Equations in which the roots are limited to certain range.
Given the number of variables , positive integers, ,  and a root limit, , create the function, numRoots(n,a,b,L), that outputs the number of posible integer root sets of the following system of equations:
                
                
        with:
                .
For example, if , the system of equations:  and , where , has only one root set, namely: . Therefore, numRoots(2,10,4,1) = 1.
If , two of the possible roots of:
                ; and
                
                where:  
are and . In fact, there are  possible root sets. Therefore, numRoots(4,20,6,3) = 16.
There are no possible roots for  and , therefore in these cases the function should return: numRoots = 0.
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NOTE: As an added challenge, only those solutions with Cody program size of less than or equal to 200 will be accepted.

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Last Solution submitted on Feb 08, 2023

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Ramon Villamangca on 7 Feb 2023

Update: Programs of cody size <= 200 is now accepted.

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Problem 57565. Easy Sequences 92: Number of Roots of a System of Linear Diophantine Equations

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Problem 57565. Easy Sequences 92: Number of Roots of a System of Linear Diophantine Equations

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Last 200 Solutions

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