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{"name": "Find $\\mu(n)$, $\\sigma(n)$, $\\tau(n)$, $\\phi(n)$", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"siga1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[(p[0]^(c[0]+1)-1)/(p[0]-1),(p[1]^(c[1]+1)-1)/(p[1]-1),(p[2]^(c[2]+1)-1)/(p[2]-1),(p[3]^(c[3]+1)-1)/(p[3]-1),(p[4]^(c[4]+1)-1)/(p[4]-1)]", "description": "", "name": "siga1"}, "sig": {"templateType": "anything", "group": "Ungrouped variables", "definition": "prodlist(siga,1,4)", "description": "", "name": "sig"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(s=0,a2+a1,a1+a2)", "description": "", "name": "a"}, "siga": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(max(x,1),x,siga1)", 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"mu1"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "name": "s"}, "mu2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "d[4]", "description": "", "name": "mu2"}, "phia1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[p[0]^(c[0]-1)*(p[0]-1),p[1]^(c[1]-1)*(p[1]-1),p[2]^(c[2]-1)*(p[2]-1),p[3]^(c[3]-1)*(p[3]-1),p[4]^(c[4]-1)*(p[4]-1)]", "description": "", "name": "phia1"}, "phia": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(max(x,1),x,phia1)", "description": "", "name": "phia"}, "notzero": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(min(x,1),x,c))", "description": "", "name": "notzero"}}, "ungrouped_variables": ["siga", "eorodd", "phia1", "notzero", "sig", "siga1", "ta", "phi", "mess", "a1", "a2", "a", "c", "b", "mu1", "d", "phia", "mu2", "n", "mu", "p", "s", "t"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find $\\mu(n)$, $\\sigma(n)$, $\\tau(n)$, $\\phi(n)$", "functions": {"prodlist": {"type": "number", "language": "jme", "definition": "if(n=-1,p,prodlist(a,p*a[n],n-1))", "parameters": [["a", "list"], ["p", "number"], ["n", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{mu}", "minValue": "{mu}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{ta}", "minValue": "{ta}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{sig}", "minValue": "{sig}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{phi}", "minValue": "{phi}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n\n\n<p>Let $n=\\var{n}$.</p>\n\n\n\n<p>Find:</p>\n\n\n\n<p>$\\mu(n) =\\;\\;$[[0]]</p>\n\n\n\n<p>$\\tau(n)=\\;\\;$[[1]]</p>\n\n\n\n<p>$\\sigma(n)=\\;\\;$[[2]]</p>\n\n\n\n<p>$\\phi(n)=\\;\\;$[[3]]</p>\n\n", "steps": [{"type": "information", "prompt": "\n\n\n<h4>Definitions.</h4>\n\n\n\n<p>$\\mu(n)$ gives information on the prime decomposition of $n$ (see formula below).</p>\n\n\n\n<p>$\\tau(n)$ gives the number of divisors of $n$.</p>\n\n\n\n<p>$\\sigma(n)$ is the sum of all divisors of $n$</p>\n\n\n\n<p>$\\phi(n)$ is the number of natural integers $\\lt n$ which are coprime to $n$.</p>\n\n\n\n<p>All are multiplicative functions i.e. $f(n)$ is multiplicative if $n_1,\\;n_2$ are coprime then:<br/>\\[ f(n_1n_2)=f(n_1)f(n_2)\\]</p>\n\n\n\n<h4>More Information and Formulae.</h4>\n\n\n\n<p>Recall that if $n=p_1^{\\beta_1}\\times p_2^{\\beta_2}\\times \\cdots \\times p_m^{\\beta_m},\\;\\;\\beta_j \\ge 1,\\;\\forall j$ is the prime factorization of $n$ then:<br/>\\[\\begin{eqnarray*}\\mu(n)&=&0, \\mbox{ if }\\exists j, p_j \\ge 2\\\\\n\n&=&(-1)^m, \\mbox{ if }\\beta_j=1,\\;\\;\\forall j\\\\\n\n\\\\\n\n\\tau(n)&=&(\\beta_1+1)\\times (\\beta_2+1) \\times\\cdots \\times (\\beta_m+1)\\\\\n\n\\\\\n\n\\sigma(n)&=& \\frac{(p_1^{\\beta_1+1}-1)}{p_1-1}\\times \\frac{(p_2^{\\beta_2+1}-1)}{p_2-1} \\times \\cdots \\times \\frac{(p_m^{\\beta_m+1}-1)}{p_m-1}\\\\\n\n\\\\\n\n\\phi(n) &=& (p_1^{\\beta_1} - p_1^{\\beta_1-1})\\times (p_2^{\\beta_2} - p_2^{\\beta_2-1})\\times\\cdots \\times (p_m^{\\beta_m} - p_m^{\\beta_m-1})\n\n\\end{eqnarray*}\n\n\\]</p>\n\n", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "", "tags": ["checked2015", "coprime", "euler totient function", "MAS3214", "multiplicative functions", "M\u00f6bius function", "number of divisors", "number theory", "prime decompositions", "sum of divisors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Number Theory.&nbsp;</p>\n<p>Given $n \\in \\mathbb{N}$ find $\\mu(n),\\;\\tau(n),\\;\\sigma(n),\\;\\phi(n).$</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\n\n<p>The factorization into a product of powers of prime factors is given by:</p>\n\n\n\n<p>$\\var{n} = \\simplify[unitFactor,unitPower,zeroPower]{{p[0]}^{c[0]}*{p[1]}^{c[1]}*{p[2]}^{c[2]}*{p[3]}^{c[3]}*{p[4]}^{c[4]}}$</p>\n\n\n\n<h4>Finding $\\mu(\\var{n})$</h4>\n\n\n\n<p>There are an {eorodd} number $\\var{notzero}$ of distinct prime factors, {mess} since the maximum power of any prime in this factorization is {d[4]} we see that</p>\n\n\n\n<p>$\\mu(\\var{n})=\\;\\var{mu}$</p>\n\n\n\n<h4>Finding $\\tau(\\var{n})$</h4>\n\n\n\n<p>Recall that if $n=p_1^{\\beta_1}\\times p_2^{\\beta_2}\\times \\cdots \\times p_m^{\\beta_m}$ is the prime factorization of $n$, then the number of divisors is given by:</p>\n\n\n\n<p>$\\tau(n)=(\\beta_1+1)\\times (\\beta_2+1) \\times\\cdots \\times (\\beta_m+1)$</p>\n\n\n\n<p>Hence in this case we have:</p>\n\n\n\n<p>$\\tau(\\var{n})=\\tau( \\simplify[unitFactor,unitPower,zeroPower]{{p[0]}^{c[0]}*{p[1]}^{c[1]}*{p[2]}^{c[2]}*{p[3]}^{c[3]}*{p[4]}^{c[4]}})=\\simplify[!basic,unitFactor,zeroTerm]{({c[0]}+1)*({c[1]}+1)*({c[2]}+1)*({c[3]}+1)*({c[4]}+1)}=\\var{ta}$</p>\n\n\n\n<h4>Finding $\\sigma(\\var{n})$</h4>\n\n\n\n<p>The sum over all divisors is given by :</p>\n\n\n\n<p>\\[\\sigma(n)= \\frac{(p_1^{\\beta_1+1}-1)}{p_1-1}\\times \\frac{(p_2^{\\beta_2+1}-1)}{p_2-1} \\times \\cdots \\times \\frac{(p_m^{\\beta_m+1}-1)}{p_m-1}\\]</p>\n\n\n\n<p>In this case we have:</p>\n\n\n\n<p>\\[\\begin{eqnarray*}\n\n\\sigma(\\var{n})&#38;=&#38;\\sigma( \\simplify[unitFactor,unitPower,zeroPower]{{p[0]}^{c[0]}*{p[1]}^{c[1]}*{p[2]}^{c[2]}*{p[3]}^{c[3]}*{p[4]}^{c[4]}})\\\\        \n\n&#38;=&#38;\\simplify[unitFactor]{{siga[0]}*{siga[1]}*{siga[2]}*{siga[3]}*{siga[4]}}\\\\\n\n&#38;=&#38;\\var{sig}\n\n\\end{eqnarray*}\n\n\\]</p>\n\n\n\n<h4>Finding $\\phi(\\var{n})$</h4>\n\n\n\n<p>This is the Euler totient function and $\\phi(n)$ it counts up the number of integers less than $n$ which are coprime to $n$.</p>\n\n\n\n<p>The formula we use for this is given by:</p>\n\n\n\n<p>\\[\\phi(n) = (p_1^{\\beta_1} - p_1^{\\beta_1-1})\\times (p_2^{\\beta_2} - p_2^{\\beta_2-1})\\times\\cdots \\times (p_m^{\\beta_m} - p_m^{\\beta_m-1})\\]</p>\n\n\n\n<p>$\\phi(\\var{n})=\\phi( \\simplify[unitFactor,unitPower,zeroPower]{{p[0]}^{c[0]}*{p[1]}^{c[1]}*{p[2]}^{c[2]}*{p[3]}^{c[3]}*{p[4]}^{c[4]}})=\\simplify[unitFactor]{{phia[0]}*{phia[1]}*{phia[2]}*{phia[3]}*{phia[4]}}=\\var{phi}$</p>\n\n", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}