Problem 52921. Easy Sequences 40: Quadratic Congruence

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>Quadratic Congruence is a modular equation of the form: </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>ax^{2}+bx+c\\equiv 0\\  (mod\\  m)</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>In this exercise you will be given a vector containing the coefficients of a quadratic polynomial (</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>cs =\\left[ a,b,c\\right]  </w:t></w:r></w:customXml><w:r><w:t>), and a modulus base (</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>m</w:t></w:r></w:customXml><w:r><w:t>). </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>Using these data, create a function that outputs the pair (</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>xs=[x_{1},x_{2}]</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>), which are the 'primitive' solutions to the quadratic congruence.</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 consider the congruence: </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^{2}+5x+6\\equiv 0\\  (mod\\  7)</w:t></w:r></w:customXml><w:r><w:t>,  the solution is </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>[4,5]</w:t></w:r></w:customXml><w:r><w:t>, since:</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>4^{2}+5\\cdot 4+6= 16+20+6= 42\\equiv 0\\  (mod\\  7)</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=\"false\"/></w:customXmlPr><w:r><w:t>5^{2}+5\\cdot 5+6= 25+25+6= 56\\equiv 0\\  (mod\\  7)</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>NOTE: </w:t></w:r><w:r><w:t>A primitive modulus to base </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>m</w:t></w:r></w:customXml><w:r><w:t>, can only have values from </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>0</w:t></w:r></w:customXml><w:r><w:t> 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>m-1</w:t></w:r></w:customXml><w:r><w:t>. This is a simplified problem, in which the quadratic polynomials given in the test suite, are all factorable, and the modulus base are all odd primes.</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: 235px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 117.5px; transform-origin: 407px 117.5px; 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; "><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: 182px 8px; transform-origin: 182px 8px; unicode-bidi: normal; "><span style="">Quadratic Congruence is a modular equation of the form: </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 167px; height: 19.5px;" width="167" height="19.5"></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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; "><span style="">. </span></span></div><div style="block-size: 64px; 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 32px; text-align: left; transform-origin: 384px 32px; 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: 297px 8px; transform-origin: 297px 8px; unicode-bidi: normal; "><span style="">In this exercise you will be given a vector containing the coefficients of a quadratic polynomial (</span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAlCAYAAACziFDNAAAEIUlEQVR4nO1c7ZGrMAzcHuiABmggFVABHdABHaSF1JAS0kNaoAZauPfD1lg4NtjGAeWNdoY5cnzYXq9kSc4doFAoFAqFQnEqBgBT4Biu7JRwhPiaANyu7NSv4QVgtj/5cb+yU8Lhc/UCsMCIT5GIF5SwGlAeM/ELhN1gln7Jy/8v8CgKUglrATxhlrA/e0jsJ0Eqj2IhnbAJTniSg3fpPIqDdMLucMKTDOk8ioN0wmYY0T2v7sgOpPMoDpIJa+G83XhxX/YgmUeRkEzYCCe87uK+7KEqjw1MQJsS1Hb2vtZ+btm5ZBwlzOeHOGuOdMriCSO6OdLmt/ltbTspoj8svMa+YAbwsOdv+zkkwAEm5afKNe0ELAD6jDZvFY6SiSghrIXh5g8m+CeO6HdLQT9CoFLKw36esC6vUPs1RE6gsc1wY1tgeNri95DwetvI7DVChL6xHiSPQfj9z8DvtnDDmszSoyQOyiWMDG3Gmoue9aNGIsA56WHm4G3Pe3teO/6j0s0T67FRgvMIPWRRLLwBzq37FkQN+5bMa0xcZDQJqZbYIbz/l3ukeliOHMK2OOJCqbHDwMsod3yKmRv9q0J75Fy22tkyqCLhcdJCy2lvX+qv9TQRoar6r2yypxLGxxry5P3O9VyQR1vseQjUXux6KsiBLAg7i9Hes+VIioRXWitq4GKOBbIr6zGkENbCjTNmUOShjoqA2uMhREjIDdI8UU5bR5bsbOFxSy4RDk/53/iNTJYjhTBahrbCBzLeGp6ez0nsfdzDHhEMH9sRZAuPEoGcmGzrHX5gmgLJWS33CLHgmgulJM70wfmMjamWYMiTH/XU2cKr0XADZ/ElnlNyVss9ekxUfOw1QHMSW0K5MRypnXXsPUc9dbbwSgLUO7YTjdxBSM5q91YEntnXiO+4GGLZ8Qt1sllu8KcLj6zVr9FxjFi7/DfC3qWGFZ6NPcJokkPCG+CK63zy7lh7ffq7jhTDiJWo/Ouhkg7sMxM+5yyElDCise/aC5+yhcdjBd+LNfa63ymytsa7N/YeydgjjNfTuHBGmMnvvOsPrJdIvhKkhAO8MOxPNi37W06CL/sxcYbuD93b2rZS6pLZwuMlEaqM01et50CjXGBkCSTQoxnWFchJLsjgZrjJ58vVgs+4bPSe9/ddfXDhkceiUCSFX39LbU80vP9PuESNtk1TnUhRHY/cKY+XYq66g1tK+DMP/L91vA5mfLFxkpcLTTIZJQl2T3iA85yc2wFp1YKePZciPMDM8509R2PJqU5I/paPSJxJ2ITt/c7a2CrH1IYKLxNnEUYlp7Pi3xHnfmtZhZeJMwhr4OLnMzDgM/n7NlR4meBxGz9qeqYz/71Dh/0N/aMI7Rq9ocLLAq/T+dmrIozYzpEKT6FQKBQKhUKhUHwX/wAL+vx42wLPWgAAAABJRU5ErkJggg==" style="width: 79px; height: 18.5px;" width="79" height="18.5"></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: 26.5px 8px; transform-origin: 26.5px 8px; unicode-bidi: normal; "><span style="">), and a modulus base (</span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">m</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: 6.5px 8px; transform-origin: 6.5px 8px; unicode-bidi: normal; "><span style="">). </span></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: 200.5px 8px; transform-origin: 200.5px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">Using these data, create a function that outputs the pair (</span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 75px; height: 20px;" width="75" height="20"></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: 56px 8px; transform-origin: 56px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">), which are the 'primitive' solutions to the quadratic congruence.</span></span></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: 123.5px 8px; transform-origin: 123.5px 8px; unicode-bidi: normal; "><span style="">For example consider the congruence: </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 157.5px; height: 19.5px;" width="157.5" height="19.5"></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: 51.5px 8px; transform-origin: 51.5px 8px; unicode-bidi: normal; "><span style="">,  the solution is </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 35px; height: 18.5px;" width="35" height="18.5"></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: 22.5px 8px; transform-origin: 22.5px 8px; unicode-bidi: normal; "><span style="">, since:</span></span></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: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; "><span style="">                                    </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 292px; height: 19.5px;" width="292" height="19.5"></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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "><span style="">, and</span></span></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: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; "><span style="">                                    </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 292px; height: 19.5px;" width="292" height="19.5"></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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "><span style="">.</span></span></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; "><span style="font-weight: 700; ">NOTE: </span></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: 88.5px 8px; transform-origin: 88.5px 8px; unicode-bidi: normal; "><span style="">A primitive modulus to base </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">m</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: 88.5px 8px; transform-origin: 88.5px 8px; unicode-bidi: normal; "><span style="">, can only have values from </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);">0</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: 10px 8px; transform-origin: 10px 8px; unicode-bidi: normal; "><span style=""> to </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 38.5px; height: 18px;" width="38.5" height="18"></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: 132px 8px; transform-origin: 132px 8px; unicode-bidi: normal; "><span style="">. This is a simplified problem, in which the quadratic polynomials given in the test suite, are all factorable, and the modulus base are all odd primes.</span></span></div></div></div>

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Problem 52921. Easy Sequences 40: Quadratic Congruence

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Problem 52921. Easy Sequences 40: Quadratic Congruence

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