// Numbas version: exam_results_page_options
{"name": "Modulo arithmetic: the least residue of a given number", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>determine the least&nbsp;residue mod n of a given number.</p>"}, "tags": [], "functions": {}, "name": "Modulo arithmetic: the least residue of a given number", "ungrouped_variables": ["a", "lr_a", "b", "lr_b", "c", "lr_c", "small_n", "d", "lr_d", "large_n", "parte", "lr_e"], "parts": [{"variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "variableReplacements": [], "minValue": "lr_a", "scripts": {}, "marks": 1, "allowFractions": false, "mustBeReduced": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "lr_a"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>The number $\\var{a}$ is congruent (mod $2$) to the least residue&nbsp;[[0]].</p>", "variableReplacements": [], "scripts": {}, "marks": 0}, {"variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "variableReplacements": [], "minValue": "lr_b", "scripts": {}, "marks": 1, "allowFractions": false, "mustBeReduced": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "lr_b"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>The number $\\var{b}$ is congruent (mod $10$) to the least residue&nbsp;[[0]].</p>", "variableReplacements": [], "scripts": {}, "marks": 0}, {"variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "variableReplacements": [], "minValue": "lr_c", "scripts": {}, "marks": 1, "allowFractions": false, "mustBeReduced": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "lr_c"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>The number $\\var{c}$ is congruent (mod $100$) to the least residue&nbsp;[[0]].</p>", "variableReplacements": [], "scripts": {}, "marks": 0}, {"variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "variableReplacements": [], "minValue": "lr_d", "scripts": {}, "marks": 1, "allowFractions": false, "mustBeReduced": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "lr_d"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>The number $\\var{d}$ is congruent (mod $\\var{small_n}$) to the least residue&nbsp;[[0]].</p>", "variableReplacements": [], "scripts": {}, "marks": 0}, {"variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "variableReplacements": [], "minValue": "lr_e", "scripts": {}, "marks": 1, "allowFractions": false, "mustBeReduced": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "lr_e"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>The number $\\var{parte}$ is congruent (mod $\\var{large_n}$) to the least residue&nbsp;[[0]].</p>", "variableReplacements": [], "scripts": {}, "marks": 0}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"lr_e": {"description": "", "definition": "mod(parte,large_n)", "templateType": "anything", "name": "lr_e", "group": "Ungrouped variables"}, "c": {"description": "", "definition": "random(10000..90000)", "templateType": "anything", "name": "c", "group": "Ungrouped variables"}, "lr_b": {"description": "", "definition": "mod(b,10)", "templateType": "anything", "name": "lr_b", "group": "Ungrouped variables"}, "lr_d": {"description": "", "definition": "mod(d,small_n)", "templateType": "anything", "name": "lr_d", "group": "Ungrouped variables"}, "b": {"description": "", "definition": "random(10000..90000)", "templateType": "anything", "name": "b", "group": "Ungrouped variables"}, "a": {"description": "", "definition": "random(2000..9000)", "templateType": "anything", "name": "a", "group": "Ungrouped variables"}, "large_n": {"description": "<p>large</p>", "definition": "random(13..99 except 20..100#10)", "templateType": "anything", "name": "large_n", "group": "Ungrouped variables"}, "parte": {"description": "", "definition": "random(3*large_n..12*large_n)", "templateType": "anything", "name": "parte", "group": "Ungrouped variables"}, "lr_a": {"description": "", "definition": "mod(a,2)", "templateType": "anything", "name": "lr_a", "group": "Ungrouped variables"}, "small_n": {"description": "", "definition": "random(3..12 except [5,10])", "templateType": "anything", "name": "small_n", "group": "Ungrouped variables"}, "lr_c": {"description": "", "definition": "mod(c,100)", "templateType": "anything", "name": "lr_c", "group": "Ungrouped variables"}, "d": {"description": "", "definition": "random(3*small_n..12*small_n)", "templateType": "anything", "name": "d", "group": "Ungrouped variables"}}, "extensions": [], "type": "question", "preamble": {"js": "", "css": ""}, "advice": "<p>One (slow!) way to determine which least residue is congruent to a number, is by repeatedly subtracting off $n$ until the result is one of the numbers from $0$ to $n-1$ (inclusive). A faster way is to divide the number by $n$ and then the <em>remainder</em> is the least residue.</p>\n<p></p>\n<p><strong>a)&nbsp;</strong>We want to determine the remainder when we divide $\\var{a}$ by $2$. Since</p>\n<p style=\"text-align: center;\">$\\var{a}\\div 2=\\var{floor(a/2)}$<span data-jme-visible=\"lr_a&gt;0\">$\\frac{\\var{lr_a}}{2}$</span></p>\n<p>the remainder, and hence the least residue, is $\\var{lr_a}$.</p>\n<p></p>\n<p>Note: In the case of mod $2$ the least residue indicates whether the number is odd or even. If the number is even the least residue will be $0$, but if the number is odd the least residue will be $1$!</p>\n<p></p>\n<p><strong>b)&nbsp;</strong>We want to determine the remainder when we divide $\\var{b}$ by $10$. Since</p>\n<p style=\"text-align: center;\">$\\var{b}\\div 10=\\var{floor(b/10)}$<span data-jme-visible=\"lr_b&gt;0\">$\\frac{\\var{lr_b}}{10}$</span></p>\n<p>the remainder, and hence the least residue, is $\\var{lr_b}$.</p>\n<p></p>\n<p>Note: In the case of mod $10$ the least residue is simply the last digit of the number!</p>\n<p></p>\n<p><strong>c)&nbsp;</strong>We want to determine the remainder when we divide $\\var{c}$ by $100$. Since</p>\n<p style=\"text-align: center;\">$\\var{c}\\div 100=\\var{floor(c/100)}$<span data-jme-visible=\"lr_c&gt;0\">$\\frac{\\var{lr_c}}{100}$</span></p>\n<p>the remainder, and hence the least residue, is $\\var{lr_c}$.</p>\n<p></p>\n<p>Note: In the case of mod $100$ the least residue is simply the last two digits of the number!</p>\n<p></p>\n<p><strong>d)&nbsp;</strong>We want to determine the remainder when we divide $\\var{d}$ by $\\var{small_n}$. Since</p>\n<p style=\"text-align: center;\">$\\var{d}\\div \\var{small_n}=\\var{floor(d/small_n)}$<span data-jme-visible=\"lr_d&gt;0\">$\\frac{\\var{lr_d}}{\\var{small_n}}$</span></p>\n<p>the remainder, and hence the least residue, is $\\var{lr_d}$.</p>\n<p></p>\n<p><strong>e)&nbsp;</strong>We want to determine the remainder when we divide $\\var{parte}$ by $\\var{large_n}$. Since</p>\n<p style=\"text-align: center;\">$\\var{parte}\\div \\var{large_n}=\\var{floor(parte/large_n)}$<span data-jme-visible=\"lr_e&gt;0\">$\\frac{\\var{lr_e}}{\\var{large_n}}$</span></p>\n<p>the remainder, and hence the least residue, is $\\var{lr_e}$.</p>", "statement": "<p>The set of integers $\\{0, 1, 2,\\ldots, n-1\\}$ is called the set of least residues modulo $n$.&nbsp;</p>", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}