// Numbas version: exam_results_page_options
{"question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "name": "Group", "questions": [{"name": "Functions: floor and roof", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Floor_function.svg.png", "/srv/numbas/media/question-resources/Floor_function.svg.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}], "tags": [], "metadata": {"description": "<p>Introduces students to the floor and ceiling functions</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>The <em>floor</em> of $x$ is defined to be the largest integer smaller than $x$. Similarly the <em>ceiling</em> of $x$ is defined to be the smallest integer bigger than $x$. We use the notation $\\lfloor x\\rfloor$ for the floor and $\\lceil x\\rceil$ for the ceiling. Evaluate the following:</p>\n<p></p>", "advice": "<p>Normally, when you round a number you round up to the nearest integer. For example $3.51$ gets rounded up to $4$, but $3.49$ gets rounded down to $3$.</p>\n<p>The floor and ceiling functions are two different ways of rounding a number.</p>\n<ul>\n<li>The floor function always rounds down: $\\lfloor 3.51\\rfloor = 3$.</li>\n<li>In contrast the ceiling function always rounds up: $\\lceil 3.49 \\rceil = 4$.</li>\n</ul>", "rulesets": {}, "variables": {"x1": {"name": "x1", "group": "Ungrouped variables", "definition": "decimal(random(1..9)+random(1..9)/10)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {"mark": {"script": "// extract question variables\nvar variables = this.question.scope.variables;\nvar unwrap = Numbas.jme.unwrapValue;\nvar x1 = unwrap(variables.x1);\n\nvar tree;\ntry {\n  tree = Numbas.jme.compile(this.studentAnswer);\n  var ans = unwrap(Numbas.jme.evaluate(tree,this.question.scope));\n    \n  if (Math.floor(x1) == ans) {\n    this.setCredit(1,\"Correct. This is the floor of $\" + x1 + \"$.\");\n  }\n  else if (Math.ceil(x1) == ans) { \n    this.setCredit(0,\"Close. This is the ceiling of $\" + x1 + \"$.\");\n  }\n  else if (ans % 1 != 0) {\n    this.setCredit(0,\"Your answer should be an integer.\"); \n  }\n  else {\n    this.setCredit(0,\"Incorrect.\"); \n  }\n}\ncatch(e) {\n    this.markingComment(e);\n}", "order": "instead"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\lfloor \\var{x1}\\rfloor$</p>", "answer": "{floor(x1)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {"mark": {"script": "// extract question variables\nvar variables = this.question.scope.variables;\nvar unwrap = Numbas.jme.unwrapValue;\nvar x1 = unwrap(variables.x1);\n\n// compute its derivative\nvar tree;\ntry {\n  tree = Numbas.jme.compile(this.studentAnswer);\n  var ans = unwrap(Numbas.jme.evaluate(tree,this.question.scope));\n  \n  if (Math.floor(x1) == ans) {\n    this.setCredit(0,\"Close. This is the floor of $\" + x1 + \"$.\");\n  }\n  else if (Math.ceil(x1) == ans) { \n    this.setCredit(1,\"Correct. This is the ceiling of $\" + x1 + \"$.\");\n  }\n  else if (ans % 1 != 0) {\n    this.setCredit(0,\"Your answer should be an integer.\"); \n  }\n  else {\n    this.setCredit(0,\"Incorrect.\"); \n  }\n}\ncatch(e) {\n    this.markingComment(e);\n}", "order": "instead"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\lceil \\var{x1}\\rceil$</p>", "answer": "{ceil(x1)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {"mark": {"script": "// extract question variables\nvar variables = this.question.scope.variables;\nvar unwrap = Numbas.jme.unwrapValue;\nvar x1 = -1*unwrap(variables.x1);\n\n// compute its derivative\nvar tree;\ntry {\n  tree = Numbas.jme.compile(this.studentAnswer);\n  var ans = unwrap(Numbas.jme.evaluate(tree,this.question.scope));\n  \n  if (Math.floor(x1) == ans) {\n    this.setCredit(1,\"Correct. This is the floor of $\" + x1 + \"$.\");\n  }\n  else if (Math.ceil(x1) == ans) { \n    this.setCredit(0,\"Close. This is the ceiling of $\" + x1 + \"$.\");\n  }\n  else if (ans % 1 != 0) {\n    this.setCredit(0,\"Your answer should be an integer.\"); \n  }\n  else {\n    this.setCredit(0,\"Incorrect.\"); \n  }\n}\ncatch(e) {\n    this.markingComment(e);\n}", "order": "instead"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "", "part": "p1", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\lfloor - \\var{x1}\\rfloor$</p>", "answer": "{floor(-x1)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {"mark": {"script": "// extract question variables\nvar variables = this.question.scope.variables;\nvar unwrap = Numbas.jme.unwrapValue;\nvar x1 = -1*unwrap(variables.x1);\n\nvar tree;\ntry {\n  tree = Numbas.jme.compile(this.studentAnswer);\n  var ans = unwrap(Numbas.jme.evaluate(tree,this.question.scope));\n  \n  if (Math.floor(x1) == ans) {\n    this.setCredit(0,\"Close. This is the floor of $\" + x1 + \"$.\");\n  }\n  else if (Math.ceil(x1) == ans) { \n    this.setCredit(1,\"Correct. This is the ceiling of $\" + x1 + \"$.\");\n  }\n  else if (ans % 1 != 0) {\n    this.setCredit(0,\"Your answer should be an integer.\"); \n  }\n  else {\n    this.setCredit(0,\"Incorrect.\"); \n  }\n}\ncatch(e) {\n    this.markingComment(e);\n}", "order": "instead"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\lceil -\\var{x1}\\rceil$</p>", "answer": "{ceil(-x1)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Functions: injective", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}], "tags": [], "metadata": {"description": "<p>Gives students an introduction to the injective property of functions</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Informally, when a function has exactly one input value for every output value we say it is <strong>injective</strong> or <strong>one-to-one</strong>.</p>\n<p>Formally, a function $f(x)$ is said to be injective when $f(x_1) = f(x_2)$ implies that $x_1=x_2$.</p>", "advice": "<p>Part a shows that to prove a function is not injective, find specific values of $x_1,x_2$ such that $f(x_1) = f(x_2)$ but $x_1 \\neq x_2$.</p>\n<p>Part b, this same method can be used to show that</p>\n<ul>\n<li>$\\sin$ is not injective since $\\sin(0) = \\sin(\\pi)$ but $0 \\neq \\pi$.</li>\n<li>$g(x)$ is not injective since $g(-1) = g(1)$ but $-1 \\neq 1$.</li>\n</ul>\n<p>There are two common methods to show a function is injective.</p>\n<ul>\n<li>If the function is continuous and has strictly positive (or strictly negative) derivative, then the function is injective. This is useful to prove that $\\tan(x)$ is injective because it is continuous on $(-\\pi/2,\\pi/2)$ and has derivative $\\sec^2(x) &gt; 0$.</li>\n<li>For some simple functions it is possible to prove the injective property directly. For example if you assume</li>\n</ul>\n<p style=\"text-align: center;\">$f(x_1) = f(x_2)$</p>\n<p>by definition of $f$</p>\n<p style=\"text-align: center;\">$2x_1 = 2x_2$</p>\n<p>and dividing by $2$:</p>\n<p style=\"text-align: center;\">$x_1=x_2$.</p>\n<p>Hence $f(x) = 2x$ is injective.</p>", "rulesets": {}, "variables": {"x1": {"name": "x1", "group": "Ungrouped variables", "definition": "decimal(random(1..9) + random(1..9)/10)", "description": "", "templateType": "anything"}, "floorx1": {"name": "floorx1", "group": "Ungrouped variables", "definition": "floor(x1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x1", "floorx1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Evaluate $\\lfloor \\var{x1} \\rfloor$ </p>", "minValue": "{floorx1}", "maxValue": "{floorx1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {"mark": {"script": "// extract question variables\nvar variables = this.question.scope.variables;\nvar unwrap = Numbas.jme.unwrapValue;\nvar x1 = unwrap(variables.x1);\nvar floorx1 = unwrap(variables.floorx1);\n\n// get the student's answers to the gaps\nvar ans = parseFloat(this.studentAnswer);\nvar floorans = Math.floor(ans);\n\nif (ans == x1) {\n    this.setCredit(0,\"You must choose a different number than $\" + x1 + \"$.\");\n} else if (floorans == floorx1) {\n    this.setCredit(1,\"Correct.\" ); \n}", "order": "instead"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find a value of $x$, other than $\\var{x1}$, such that $f(x) = \\var{floorx1}$</p>", "minValue": "{floorx1}", "maxValue": "{floorx1 + 0.9999999}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Hence $\\lfloor \\,\\, \\rfloor : \\mathbb R \\mapsto \\mathbb Z$ is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Injective", "Not Injective"], "matrix": [0, "1"], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Functions: surjective", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}], "functions": {}, "variable_groups": [], "advice": "<p>This question introduces the surjective property of a function. You will need to be able to confidently identify the domain, codomain and range of a function before you can understand the surjective property.</p>\n<p>There are two proof styles.</p>\n<ul>\n<li>To prove that a function $f$ is surjective, start with a generic element $b$ from the codomain and, by solving $f(a) = b$ for $a$, show that $a$ is in the domain.</li>\n<li>Always prove that a function $f$ is not surjective by contradiction. Start with a specific element $b \\in B$ and, by solving $f(a) = b$ for $a$, show that $a$ is not in the domain. This contradiction proves it is not possible for the function to be surjective.</li>\n</ul>\n<hr/>\n<p>The codomain of $f_1$ is $\\mathbb Z$. Suppose that $f_1$ is surjective, which means that the whole codomain gets mapped to, and in particular that there is some integer $a$ such that $f(a) = 1 \\in \\mathbb Z$. Then</p>\n<p style=\"text-align: center;\">$f_1(a) = 2a = 1$</p>\n<p style=\"text-align: left;\">and so $a = \\frac{1}{2}$. But $\\frac{1}{2} \\not\\in \\operatorname{Domain}(f_1) = \\mathbb Z$, so our assumption that $f_1$ is surjective must be false.</p>\n<hr/>\n<p style=\"text-align: left;\">The codomain of $f_2$ is $\\mathbb Z$. For any $b \\in \\mathbb Z$ suppose that $f(a) = b$. Then</p>\n<p style=\"text-align: center;\">$f_2(a) = \\lceil a \\rceil = b$.</p>\n<p style=\"text-align: left;\">There are many values of $a$ which map to $b$, and one such value is just $a=b \\in \\operatorname{Domain}(f_2) = \\mathbb R$. So $f_2$ is surjective.</p>\n<hr/>\n<p style=\"text-align: left;\">The codomain of $f_3$ is the positive rationals $\\mathbb Q^+$. For any $\\frac{p}{q} \\in \\mathbb Q^+$ suppose that $f(a) = \\frac{p}{q}$. Then</p>\n<p style=\"text-align: center;\">$f_3(a) = \\frac{1}{a} = \\frac{p}{q}$</p>\n<p style=\"text-align: left;\">and so $a = \\frac{q}{p} \\in \\operatorname{Domain}(f_3) = \\mathbb Q^+$. So $f_3$ is surjective.</p>", "ungrouped_variables": [], "variables": {}, "tags": [], "statement": "<p>Consider the example</p>\n<p style=\"text-align: center;\">$f: \\mathbb Z \\mapsto \\mathbb R, \\quad f(x) = \\left|x\\right|$.</p>\n<ul>\n<li style=\"text-align: left;\">The set of possible inputs $\\mathbb Z$ is called the <strong>domain</strong>.</li>\n<li style=\"text-align: left;\">The set of possible outputs $\\mathbb R$ is called the <strong>codomain</strong>.</li>\n</ul>\n<p>A function must be defined for every element of its domain, but the codomain may contain additional elements that are unused.</p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution-ShareAlike 4.0 International", "description": "<p>An introduction to terminology about the surjective property of a function.</p>"}, "parts": [{"showFeedbackIcon": true, "maxMarks": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "shuffleChoices": true, "choices": ["<p>$\\mathbb N$</p>", "<p>$\\mathbb Z^+$</p>", "<p>$\\mathbb Z$</p>", "<p>$\\mathbb R$</p>"], "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "displayType": "radiogroup", "minMarks": 0, "matrix": ["1", 0, 0, 0], "distractors": ["", "$f(0) = 0 \\not\\in \\mathbb Z^+$, and the range must include all the elements that are mapped to.", "There is no integer $a$ such that $f(a) = a^2 = -1 \\in \\mathbb Z$.", "There is no integer $a$ such that $f(a) = a^2 = \\sqrt{2} \\in \\mathbb R$."], "displayColumns": 0, "type": "1_n_2", "prompt": "<p><strong></strong>The set of <em>actual</em> outputs is called the <strong>range</strong> and may be different to the codomain. In other words, the range is the set of all elements that are actually mapped to, or in the language of sets </p>\n<p style=\"text-align: center;\">$\\operatorname{Range}(f) = \\left\\{ b | \\, f(a) = b, a \\in \\mathbb Z\\right\\}$.</p>\n<p>What is the range of $f$?</p>"}, {"showFeedbackIcon": true, "maxMarks": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "shuffleChoices": true, "choices": ["<p>$\\mathbb Z$</p>", "<p>$\\mathbb N$</p>", "<p>$\\mathbb R$</p>"], "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "displayType": "radiogroup", "minMarks": 0, "matrix": [0, 0, "1"], "distractors": ["The question asked for codomain, but you selected the domain.", "The question asked for codomain, but you selected the range.", ""], "displayColumns": 0, "type": "1_n_2", "prompt": "<p>If the codomain and range are equal the function is said to be <strong>surjective</strong> or <strong>onto</strong>. The function $f$ is not surjective because the codomain is different to the range. The codomain is</p>"}, {"showFeedbackIcon": true, "maxMarks": 0, "maxAnswers": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "shuffleChoices": false, "choices": ["<p>$f_1 : \\mathbb Z \\mapsto \\mathbb Z, \\quad f_1(x) = 2x$</p>", "<p>$f_2 : \\mathbb R \\mapsto \\mathbb Z, \\quad f_2(x) = \\lceil x\\rceil$</p>", "<p>$f_3 : \\mathbb Q^+ \\mapsto \\mathbb Q^+, \\quad f_3(x) = \\frac{1}{x}$</p>"], "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "distractors": ["The range of $f_1$ is the even integers, which is not the same as the codomain.", "", ""], "displayType": "checkbox", "minMarks": 0, "matrix": ["-1", "1", "1"], "minAnswers": 0, "displayColumns": 0, "type": "m_n_2", "prompt": "<p>Select all functions that are surjective. For this question the set of positive rational numbers is denoted by $\\mathbb Q^+$.</p>", "warningType": "none"}], "preamble": {"js": "", "css": ""}, "rulesets": {}, "type": "question"}, {"name": "Functions: bijective", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}], "parts": [{"variableReplacements": [], "gaps": [{"variableReplacements": [], "checkVariableNames": false, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "showFeedbackIcon": true, "showPreview": true, "unitTests": [], "vsetRangePoints": 5, "marks": 1, "failureRate": 1, "customMarkingAlgorithm": "", "expectedVariableNames": [], "scripts": {}, "answer": "1/b"}], "prompt": "<p><strong>Prove that $f(x)$ surjective.</strong></p>\n<p>Assume that $b \\in \\mathbb R$. We must find some $a$ such that $f(a) = b$. There are two cases to consider.</p>\n<p>If $b=0$ then choose $a = 0$.</p>\n<p>Otherwise, $b \\neq 0$ and then choose $a = $ [[0]].</p>\n<p>In either case, $f(a) = b$ and so the function is surjective.</p>", "extendBaseMarkingAlgorithm": true, "type": "gapfill", "sortAnswers": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "variableReplacementStrategy": "originalfirst"}, {"variableReplacements": [], "gaps": [{"variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "maxValue": "0", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "unitTests": [], "marks": 1, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "allowFractions": false, "minValue": "0", "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "prompt": "<p><strong>Prove that $f(x)$ is injective.</strong></p>\n<p>Assume that $f(x_0) = f(x_1)$. We must prove that $x_0 = x_1$. There are two cases to consider.</p>\n<p>If $f(x_0) = f(x_1) = 0$ then $x_0 = x_1 = $ [[0]].</p>\n<p>Otherwise, $f(x_0) = f(x_1) \\neq 0$ and then</p>\n<p style=\"text-align: center;\">$x_0 = \\frac{1}{f(x_0)} = \\frac{1}{f(x_1)} = x_1$.</p>\n<p style=\"text-align: left;\">In either case, $x_0 = x_1$ and so the function is injective.</p>", "extendBaseMarkingAlgorithm": true, "type": "gapfill", "sortAnswers": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "variableReplacementStrategy": "originalfirst"}, {"variableReplacements": [], "gaps": [{"variableReplacements": [], "checkVariableNames": false, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "showFeedbackIcon": true, "showPreview": true, "unitTests": [], "vsetRangePoints": 5, "marks": 1, "failureRate": 1, "customMarkingAlgorithm": "", "expectedVariableNames": [], "scripts": {}, "answer": "x"}], "prompt": "<p>Because $f$ is surjective and injective, it is bijective and therefore has an inverse.</p>\n<p>In this case we can actually see that $f$ is its own inverse because $f(f(x)) = $ [[0]].</p>", "extendBaseMarkingAlgorithm": true, "type": "gapfill", "sortAnswers": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "tags": [], "statement": "<p>A function which is both surjective and injective is said to be <strong>bijective</strong>.</p>\n<p>This is an important property because it guarantees that $f: A \\mapsto B$ has an inverse function $f^{-1} : B \\mapsto A$.</p>\n<hr/>\n<p>Let's prove that the function $f: \\mathbb R \\mapsto \\mathbb R$ defined by</p>\n<p style=\"text-align: center;\">$ f(x) = \\left\\{ \\begin{array}{cl} \\frac{1}x &amp; x \\neq 0 \\\\ 0 &amp; x = 0.\\end{array}\\right.$</p>\n<p style=\"text-align: left;\">has an inverse.</p>\n<p style=\"text-align: center;\"></p>", "advice": "<p>This question summarizes the definitions of surjective and injective, and applies them to prove the existance of an inverse.</p>\n<hr/>\n<p>For continuous functions it is possible to prove that it injective by using the derivative. But $f$ is not continuous, and nor is it strictly increasing or strictly decreasing. In fact it is strictly decreasing for $x&lt;0$ and strictly increasing for $x&gt;0$.</p>\n<p style=\"text-align: center;\">{myscript()}</p>\n<p style=\"text-align: left;\">So we pretty much have to prove that $f$ is injective directly from the definition.</p>\n<hr/>\n<p style=\"text-align: left;\">Since $f$ is a bijection we know that an inverse exists, but this information does not tell us how to find it. This example is special because the inverse is easy to find.</p>", "preamble": {"css": "", "js": ""}, "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution-ShareAlike 4.0 International", "description": "<p>This question summarizes the definitions of surjective and injective, and applies them to prove the existance of an inverse.</p>"}, "functions": {"myscript": {"definition": "// First, make the JSXGraph board.\n// The function provided by the JSXGraph extension wraps the board up in\n// a div tag so that it's easier to embed in the page.\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n    {boundingBox: [-4,4,4,-4],\n     axis: true,\n     showNavigation: false,\n     grid: true\n    });\n\n// div.board is the object created by JSXGraph, which you use to\n// manipulate elements\nvar board = div.board;\n\n// Then do whatever you want with the board....\nboard.create('functiongraph',[function(x){ return (0 == x ? 0 : 1/x);},0.01,4]);\nboard.create('functiongraph',[function(x){ return (0 == x ? 0 : 1/x);},-4,0.01]);\nvar p = board.createElement('point',[0,0], {visible: true, size:1, color:'blue',fixed:true});\np.setLabel(\"\");\n\n//var theLine = board.create('line',[[-4,0],[4,0]],{visible: false});\n//var x = board.create('glider',[theLine],{name:'x'});\n//var pointoncurve = board.create('point',[x.X(),(0 == x.X() ? 0 : 1/x.X())],{name:'f2',visible:true,size:3,color:'blue'});\n\n\n// and return the container div\nreturn div;", "type": "html", "language": "javascript", "parameters": []}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [], "ungrouped_variables": [], "variables": {}, "type": "question"}, {"name": "Modular Arithmetic: time", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}], "ungrouped_variables": ["hours24", "hours12", "start", "duration", "ansA", "ansB", "startb", "ansC", "c"], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a. In 24 hour time: {hours24[start]} + {duration}:00 = {start+duration+1}:00 = {hours24[mod(start+duration,24)]}</p>\n<p>b. In 12 hour time {hours12[startb]} + {duration}:00 = {startb+duration+1}:00 = {hours12[mod(startb+duration,12)]}</p>\n<p>c. A time of {hours12[c]} in 12-hour format could mean {hours24[c]} or {hours24[c+12]} in 24-hour format.</p>", "metadata": {"description": "<p>Simple intro to mod arithmetic</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "rulesets": {}, "functions": {}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "tags": [], "variables": {"startb": {"group": "Ungrouped variables", "name": "startb", "definition": "random(5..10)", "templateType": "anything", "description": ""}, "c": {"group": "Ungrouped variables", "name": "c", "definition": "random(0..11)", "templateType": "anything", "description": ""}, "hours24": {"group": "Ungrouped variables", "name": "hours24", "definition": "[\"01:00\",\"02:00\",\"03:00\",\"04:00\",\"05:00\",\"06:00\",\"07:00\",\"08:00\",\"09:00\",\"10:00\",\"11:00\",\"12:00\",\"13:00\",\"14:00\",\"15:00\",\"16:00\",\"17:00\",\"18:00\",\"19:00\",\"20:00\",\"21:00\",\"22:00\",\"23:00\",\"00:00\"]", "templateType": "anything", "description": ""}, "start": {"group": "Ungrouped variables", "name": "start", "definition": "random(18..22 except startb+12)", "templateType": "anything", "description": ""}, "hours12": {"group": "Ungrouped variables", "name": "hours12", "definition": "[\"1:00\",\"2:00\",\"3:00\",\"4:00\",\"5:00\",\"6:00\",\"7:00\",\"8:00\",\"9:00\",\"10:00\",\"11:00\",\"12:00\"]", "templateType": "anything", "description": ""}, "ansC": {"group": "Ungrouped variables", "name": "ansC", "definition": "map(if(mod(x,12) = c,1,-1), x, 0..23)", "templateType": "anything", "description": ""}, "ansB": {"group": "Ungrouped variables", "name": "ansB", "definition": "map(if(mod(startb + duration,12) = x,1,0), x, 0..11)", "templateType": "anything", "description": ""}, "ansA": {"group": "Ungrouped variables", "name": "ansA", "definition": "map(if(mod(start + duration,24) = x,1,0), x, 0..23)", "templateType": "anything", "description": ""}, "duration": {"group": "Ungrouped variables", "name": "duration", "definition": "random(7..10)", "templateType": "anything", "description": ""}}, "parts": [{"showCorrectAnswer": true, "maxMarks": 0, "customName": "", "type": "1_n_2", "variableReplacements": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "displayType": "dropdownlist", "matrix": "{ansA}", "shuffleChoices": false, "minMarks": 0, "scripts": {}, "useCustomName": false, "displayColumns": 0, "variableReplacementStrategy": "originalfirst", "prompt": "<p>If you go to bed at $\\var{hours24[start]}$ and sleep for $\\var{duration}$ hours, at what time will you wake up? Give your answer in 24-hour format.</p>", "showCellAnswerState": true, "choices": "{hours24}", "marks": 0}, {"showCorrectAnswer": true, "maxMarks": 0, "customName": "", "type": "1_n_2", "variableReplacements": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "displayType": "dropdownlist", "matrix": "{ansB}", "shuffleChoices": false, "minMarks": 0, "scripts": {}, "useCustomName": false, "displayColumns": 0, "variableReplacementStrategy": "originalfirst", "prompt": "<p>If you go to bed at $\\var{hours12[startb]}$ and sleep for $\\var{duration}$ hours, at what time will you wake up? Give your answer in 12-hour format.</p>", "showCellAnswerState": true, "choices": "{hours12}", "marks": 0}, {"displayColumns": 0, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "displayType": "checkbox", "useCustomName": false, "minMarks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "<p>Deep in thought about discrete mathematics, you lost track of day and night. You glance at a 12-hour clock that reads $\\var{hours12[c]}$. What time is it in 24-hour time (tick all that could be true)?</p>", "minAnswers": 0, "maxAnswers": 0, "showCorrectAnswer": true, "customName": "", "warningType": "none", "variableReplacements": [], "maxMarks": 0, "shuffleChoices": false, "matrix": "{ansC}", "scripts": {}, "type": "m_n_2", "showCellAnswerState": true, "choices": "{hours24}", "marks": 0}], "statement": "<p>The concept of modular arithmetic is already familiar from everyday life.</p>", "type": "question"}, {"name": "Modular Arithmetic: quotient and remainder", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}], "tags": [], "metadata": {"description": "<p>Gentle intro to modular arithmetic through quotients and remainders</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Given integers $a,b \\in \\mathbb Z$ we can write $a$ in the form</p>\n<p style=\"text-align: center;\">$ a = qb + r$</p>\n<p style=\"text-align: left;\">for some $q,r \\in \\mathbb Z$ where $0 \\leq r &lt; b$. The numbers $q$ and $r$ are called the <strong>quotient</strong> and <strong>remainder</strong> when $a$ is divided by $b$:</p>\n<p style=\"text-align: center;\">$\\displaystyle \\frac{a}{b} = q + \\frac{r}{b}$.</p>\n<p>In <strong>modular arithmetic</strong> (in this case, modulo $b$) our interest is focused on the remainder. The operation $a \\operatorname{mod} b = r$.</p>", "advice": "<p>The first two questions introduce the idea that different numbers in $\\mathbb Z$ can be the same in modulo $b$. For example in modulo $\\var{p}$ the numbers</p>\n<p style=\"text-align: center;\">$\\var{p}, \\var{2p}, \\var{3p}, \\cdots$</p>\n<p style=\"text-align: left;\">are equivalent to zero. While the numbers</p>\n<p style=\"text-align: center;\">$\\var{p2+1}, \\var{2p2+1}, \\var{3p2+1}, \\cdots$</p>\n<p style=\"text-align: left;\">are equivalent to $1 \\pmod{\\var{p2}}$.</p>\n<hr/>\n<p style=\"text-align: left;\">The next question introduce a simple divisibility test: $a=qb = qb+0$ exactly when $a\\equiv 0\\pmod{b}$. This is just another way to say that $b$ divides $a$, or that $a$ is a multiple of $b$. In particular</p>\n<p style=\"text-align: center;\">$\\var{p*p2} = \\var{p}\\times \\var{p2}$ is divisible by $\\var{p}$ and $\\var{p2}$.</p>", "rulesets": {}, "variables": {"p2": {"name": "p2", "group": "Ungrouped variables", "definition": "random(4..9 except p)", "description": "", "templateType": "anything"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(4..9)", "description": "", "templateType": "anything"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "random(1..(p-2))", "description": "", "templateType": "anything"}, "r2": {"name": "r2", "group": "Ungrouped variables", "definition": "random(1..(p2-2) except r)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["p", "r", "p2", "r2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Which numbers have a remainder of $\\var{r}$ when divided by $\\var{p}$? These numbers would be considered equal in modulo $\\var{p}$.</p>", "minMarks": 0, "maxMarks": "1", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "choices": ["<p>$\\var{r+p}$</p>", "<p>$\\var{r-p}$</p>", "<p>$\\var{r+p+1}$</p>", "<p>$\\var{r-p-1}$</p>", "<p>$\\var{r-2p+1}$</p>", "<p>$\\var{r-2p}$</p>", "<p>$\\var{r+3p}$</p>"], "matrix": ["0.25", "0.25", "-0.25", "-0.25", "-0.25", "0.25", "0.25"], "distractors": ["$\\var{r+p} = 1\\times \\var{p} + \\var{r}$ and so has remainder $\\var{r}$.", "$\\var{r-p} = (-1)\\times \\var{p} + \\var{r}$ and so has remainder $\\var{r}$.", "$\\var{r+p+1} = 1\\times \\var{p} + \\var{r+1}$ and so has remainder $\\var{r+1}$.", "$\\var{r-p-1} = (-1)\\times \\var{p} + \\var{r-1}$ and so has remainder $\\var{r-1}$.", "$\\var{r-2p-1} = (-2)\\times \\var{p} + \\var{r-1}$ and so has remainder $\\var{r-1}$.", "$\\var{r-2p} = (-2)\\times \\var{p} + \\var{r}$ and so has remainder $\\var{r}$.", "$\\var{r+3p} = 3\\times \\var{p} + \\var{r}$ and so has remainder $\\var{r}$."]}, {"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Which numbers are equal to $\\var{r2}$ in modulo $\\var{p2}$?</p>", "minMarks": 0, "maxMarks": "1", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "choices": ["<p>$\\var{r2+p2}$</p>", "<p>$\\var{r2-p2}$</p>", "<p>$\\var{r2+p2+1}$</p>", "<p>$\\var{r2-p2-1}$</p>", "<p>$\\var{r2}$</p>", "<p>$\\var{r2-2*p2}$</p>", "<p>$\\var{r2+3*p2-1}$</p>"], "matrix": ["0.25", "0.25", "-0.25", "-0.25", "0.25", "0.25", "-0.25"], "distractors": ["$\\var{r2+p2} = 1\\times \\var{p2} + \\var{r2}$ and so has remainder $\\var{r2}$.", "$\\var{r2-p2} = (-1)\\times \\var{p2} + \\var{r2}$ and so has remainder $\\var{r2}$.", "$\\var{r2+p2+1} = 1\\times \\var{p2} + \\var{r2+1}$ and so has remainder $\\var{r2+1}$.", "$\\var{r2-p2-1} = (-1)\\times \\var{p2} + \\var{r2-1}$ and so has remainder $\\var{r2-1}$.", "$\\var{r2} = (0)\\times \\var{p2} + \\var{r2}$ and so has remainder $\\var{r2}$.", "$\\var{r2-2*p2} = (-2)\\times \\var{p2} + \\var{r2}$ and so has remainder $\\var{r2}$.", "$\\var{r2+3*p2-1} = 3\\times \\var{p2} + \\var{r2-1}$ and so has remainder $\\var{r2-1}$."]}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {"mark": {"script": "// extract question variables\nvar variables = this.question.scope.variables;\nvar unwrap = Numbas.jme.unwrapValue;\nvar p1 = unwrap(variables.p);\nvar p2 = unwrap(variables.p2);\n\ntry {\n  // get the student's answers to the first two steps\n  var ans = Numbas.util.parseNumber(this.studentAnswer);\n  \n  var grade = 0;\n  \n  if (ans == 0) {\n      this.markingComment(\"The question asks for a non-zero number.\");\n    }\n  else {\n    var p = p1;\n    for (var i = 0; i < 2; i++) \n    { \n      if (0 == (ans % p)) {\n        grade += 0.5;\n        this.markingComment(\"$\" + ans + \" = \" + (ans % p) + \" \\\\pmod{\" + p + \"}$.\");\n      } \n      else {\n        this.markingComment(\"$\" + ans + \" = \" + (ans % p) + \" \\\\neq 0 \\\\pmod{\" + p + \"}$.\");\n      };\n      p = p2;\n    }\n  }\n  this.setCredit(grade,\"\");\n  //var ans = unwrap(Numbas.jme.evaluate(tree,this.question.scope));\n  \n}\ncatch(e) {\n    this.markingComment(e);\n}", "order": "instead"}, "validate": {"script": "return Numbas.util.isNumber(this.studentAnswer,false,\"\");", "order": "instead"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A number $a$ is a <strong>multiple</strong> of $b$ precisely when $a=qb$ for some number $q$. In terms of modular arithmetic we would say $a \\operatorname{mod} b = 0$ because $a$ has no remainder when divided by $b$.</p>\n<p>Enter a non-zero number which is equivalent to $0 \\pmod{\\var{p}}$ and also equivalent to $0 \\pmod{\\var{p2}}$.</p>", "minValue": "{p*p2}", "maxValue": "{p*p2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Modular Arithmetic: Greatest Common Divisor", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}], "tags": [], "metadata": {"description": "<p>Introduces the GCD and a simple form of the Euclidian Algorithm</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>A number $n$ <strong>divides</strong> $a$ when $a = nq$ for some $q$.</p>\n<p>Any number which divides both $a$ and $b$ is called a <strong>common d</strong><strong>ivisor</strong> of $a$ and $b$, the largest common divisor is called the <strong>greatest common divisor</strong> and is denoted $\\gcd(a,b)$.</p>", "advice": "<p>Every number divides zero, so the largest number to divide $\\var{a1}$ is the greatest common divisor of $0$ and $\\var{a1}$. This means that $\\gcd(\\var{a1},0) = \\var{a1}$.</p>\n<hr/>\n<p>The Euclidian Algorithm presented here is a variation on the usual presentation. Another common method keeps track of the quotients:</p>\n<p style=\"text-align: center;\">$ \\var{gcd*(1+b*(1+a))} = \\var{b}\\times (\\var{gcd*(1+a)}) + \\var{a1} $</p>\n<p style=\"text-align: center;\">$ \\var{gcd*(1+a)} = \\var{gcd*(1+a)/a1}\\times (\\var{a1}) + 0 $</p>\n<p>to write the $\\gcd$ in the form of <a href=\"https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity\" target=\"_blank\" rel=\"noopener noreferrer\" title=\"B&eacute;zout's identity\">B&eacute;zout's identity</a></p>\n<p style=\"text-align: center;\">$ \\var{gcd*(1+b*(1+a))} - \\var{b}\\times (\\var{gcd*(1+a)}) = \\var{a1}$.</p>", "rulesets": {}, "variables": {"a2": {"name": "a2", "group": "Ungrouped variables", "definition": "mod(gcd*(a+1),a1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..9 except [a,gcd])", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "mod(gcd*(1+b*(1+a)),gcd*(1+a))", "description": "", "templateType": "anything"}, "big": {"name": "big", "group": "Ungrouped variables", "definition": "random(100..199)", "description": "", "templateType": "anything"}, "gcd": {"name": "gcd", "group": "Ungrouped variables", "definition": "random(3..19)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9 except gcd)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "gcd*(1+a)/a1", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "gcd", "a1", "a2", "c", "big"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>As a consequence of these definitions, every number $n$ divides zero because $0 = n\\times 0$.</p>\n<p>So $\\gcd(\\var{big},0)$ is</p>", "minValue": "{big}", "maxValue": "{big}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>What is $\\gcd(\\var{gcd*(1+b*(1+a))},\\var{gcd*(1+a)})$?</p>", "stepsPenalty": 0, "steps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>The procedure is to use the identity</p>\n<p style=\"text-align: center;\">&nbsp;$\\gcd(a,b) = \\gcd(a\\mod b, b)$.</p>\n<p style=\"text-align: left;\">to simplify the question.</p>\n<p>Compute $\\var{gcd*(1+b*(1+a))} \\mod{\\var{gcd*(1+a)}}$.</p>", "minValue": "{a1}", "maxValue": "{a1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Hence $\\gcd(\\var{gcd*(1+b*(1+a))},\\var{gcd*(1+a)}) = \\gcd(\\var{a1},\\var{gcd*(1+a)})$. We can simplify this again using the identity</p>\n<p style=\"text-align: center;\">&nbsp;$\\gcd(a,b) = \\gcd(a, b\\mod a)$.</p>\n<p style=\"text-align: left;\">Compute $\\var{gcd*(a+1)} \\mod{\\var{a1}} $.</p>", "minValue": "{a2}", "maxValue": "{a2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p style=\"text-align: left;\">By re-applying these identities, the question has been simplified to something managable:</p>\n<p style=\"text-align: center;\">$\\gcd(\\var{gcd*(1+b*(1+a))},\\var{gcd*(1+a)}) = \\gcd(\\var{a1},\\var{gcd*(1+a)}) = \\gcd(\\var{a1}, 0)$.</p>"}], "minValue": "{gcd}", "maxValue": "{gcd}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}]}]}], "feedback": {"showactualmark": true, "allowrevealanswer": true, "intro": "", "showtotalmark": true, "showanswerstate": true, "feedbackmessages": [], "advicethreshold": 0}, "showstudentname": true, "name": "Week 2: Functions and modular arithmetic", "metadata": {"licence": "Creative Commons Attribution-ShareAlike 4.0 International", "description": "<p>Formative assessment to introduce the concepts of modular arithmetic.</p>"}, "percentPass": 0, "showQuestionGroupNames": false, "navigation": {"showfrontpage": true, "allowregen": true, "onleave": {"message": "", "action": "none"}, "reverse": true, "showresultspage": "oncompletion", "preventleave": true, "startpassword": "", "browse": true}, "timing": {"allowPause": true, "timedwarning": {"message": "", "action": "none"}, "timeout": {"message": "", "action": "none"}}, "duration": 0, "type": "exam", "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}], "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/Floor_function.svg.png", "/srv/numbas/media/question-resources/Floor_function.svg.png"]]}