// Numbas version: finer_feedback_settings
{"metadata": {"description": "<p>Given two numbers, find the gcd, then use B&eacute;zout's algorithm to find $s$ and $t$ such that $as+bt=\\operatorname{gcd}(a,b)$.</p>\n<p>Finally, find all solutions of an equation $\\mod b$.</p>", "licence": "Creative Commons Attribution 4.0 International", "notes": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "pickQuestions": 0, "questions": [{"name": "Greatest common divisor and B\u00e9zout's algorithm", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "ungrouped_variables": ["s", "diff", "b1", "b", "n", "t", "a1", "bb", "c", "describeleast", "a", "least", "d", "k"], "variable_groups": [], "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given two numbers, find the gcd, then use B&eacute;zout's algorithm to find $s$ and $t$ such that $as+bt=\\operatorname{gcd}(a,b)$. Finally, find all solutions of an equation $\\mod b$.</p>"}, "statement": "", "tags": ["euclidean algorithm", "gcd", "greatest common divisor", "Number theory", "number theory"], "parts": [{"type": "gapfill", "variableReplacements": [], "prompt": "\n\n\n<p>Find $d = \\operatorname{gcd}(\\var{a},\\var{b})$, the greatest common divisor of $\\var{a}$ and $\\var{b}$:</p>\n\n\n\n<p>$d = \\phantom{{}}$[[0]]</p>\n\n\n\n<p>Now find integers $s$ and $t$ such that:</p>\n\n\n\n<p>\\[\\var{a}s+\\var{b}t = d\\]</p>\n\n\n\n<p>$s = \\phantom{{}}$[[1]] $, t = \\phantom{{}}$[[2]]$.$</p>\n\n\n  ", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "gaps": [{"correctAnswerStyle": "plain", "type": "numberentry", "minValue": "d", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "d", "scripts": {}, "marks": 1, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false}, {"correctAnswerStyle": "plain", "type": "numberentry", "minValue": "s", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "s", "scripts": {}, "marks": 1, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false}, {"correctAnswerStyle": "plain", "type": "numberentry", "minValue": "t", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "t", "scripts": {}, "marks": 1, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false}], "showFeedbackIcon": true, "scripts": {}}, {"type": "gapfill", "variableReplacements": [], "prompt": "\n\n\n<p>Find all solutions $x$ of the following congruence:</p>\n\n\n\n<p>\\[\\var{a}x \\equiv \\var{n} \\mod \\var{b} \\]</p>\n\n\n\n<p>The least solution $x$ such that $0 \\lt x \\lt \\var{b}$ is: [[0]]</p>\n\n\n\n<p>What is the difference between two successive solutions? [[1]]</p>\n\n\n  ", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "gaps": [{"correctAnswerStyle": "plain", "type": "numberentry", "minValue": "{least}", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "{least}", "scripts": {}, "marks": 1, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false}, {"correctAnswerStyle": "plain", "type": "numberentry", "minValue": "{diff}", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "{diff}", "scripts": {}, "marks": 1, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerFraction": false}], "showFeedbackIcon": true, "scripts": {}}], "advice": "<h4>a)</h4>\n<p>On applying the standard method for finding the $\\operatorname{gcd}$ of two numbers we have the following sequence:</p>\n<p>{describegcd(a,b)}</p>\n<p>The last non-zero remainder is $\\var{d}$, so this is the $\\operatorname{gcd}$ of $\\var{a}$ and $\\var{b}$.</p>\n<p>Now work backwards through those steps, rearranging them to find the remainders as linear combinations of the other numbers.</p>\n<p>When you reach the last line you will have found $s$ and $t$ such that <br/>\\[\\var{a}s+\\var{b}t = \\var{d}\\]</p>\n<p>{describebezout(a,b)}</p>\n<p>So $s = \\var{s}$ and $t = \\var{t}$.</p>\n<hr/>\n<h4>b)</h4>\n<h3 style=\"text-align: center;\">First Step</h3>\n<p>We would like to find all the solutions $x$ of the equation<br/>\\[\\var{a}x \\equiv \\var{n} \\mod \\var{b}.\\]</p>\n<p>In order to solve such an equation first determine if there is a solution at all.</p>\n<hr/>\n<h5>Lemma</h5>\n<p>An equation of the form $ax \\equiv n \\mod b$ has a solution if and only if $\\operatorname{gcd}(a,b)|n$.</p>\n<hr/>\n<p>So the first task is to find $\\operatorname{gcd}(a,b)$ and see if it divides $n$. If not, there is no solution.</p>\n<p>In the first part, we found that $d = \\operatorname{gcd}(\\var{a},\\var{b}) = \\var{d}$.</p>\n<p>$\\var{n} = \\var{d} \\times \\var{k}$, so the equation $\\var{a}x \\equiv \\var{n} \\mod \\var{b}$ has a solution.</p>\n<h3 style=\"text-align: center;\">Finding a solution</h3>\n<p>Note that because $d|n$, then $n=kd$ for some integer $k$.</p>\n<p>In order to find a solution of $x$ go back to the first part and use the integers $s$ and $t$ such that</p>\n<p>\\[ \\var{a}s + \\var{b}t = d. \\]</p>\n<p>If we multiply both sides of this equation by $k$ we find that</p>\n<p>\\[ \\var{a}ks + \\var{b}kt = kd = n, \\]</p>\n<p>and by taking both sides modulo $\\var{b}$ we see that</p>\n<p>\\[ \\var{a}ks \\equiv n \\mod \\var{b}. \\]</p>\n<p>So $x = ks = \\var{k*s}$ is a solution of the equation.</p>\n<p>The difference between successive terms is $\\frac{\\var{b}}{\\var{d}} = \\var{diff}$.</p>\n<p>{describeLeast}</p>", "variablesTest": {"maxRuns": 100, 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