103 lines
2.7 KiB
Markdown
103 lines
2.7 KiB
Markdown
---
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id: 5900f38d1000cf542c50fea0
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title: 'Problem 33: Digit cancelling fractions'
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challengeType: 5
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forumTopicId: 301987
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dashedName: problem-33-digit-cancelling-fractions
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---
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# --description--
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The fraction <sup>49</sup>/<sub>98</sub> is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that <sup>49</sup>/<sub>98</sub> = <sup>4</sup>/<sub>8</sub>, which is correct, is obtained by cancelling the 9s.
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We shall consider fractions like, <sup>30</sup>/<sub>50</sub> = <sup>3</sup>/<sub>5</sub>, to be trivial examples.
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There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.
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If the product of these four fractions is given in its lowest common terms, find the value of the denominator.
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# --hints--
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`digitCancellingFractions()` should return a number.
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```js
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assert(typeof digitCancellingFractions() === 'number');
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```
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`digitCancellingFractions()` should return 100.
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```js
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assert.strictEqual(digitCancellingFractions(), 100);
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```
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# --seed--
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## --seed-contents--
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```js
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function digitCancellingFractions() {
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return true;
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}
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digitCancellingFractions();
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```
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# --solutions--
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```js
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function digitCancellingFractions() {
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function isCurious(numerator, denominator) {
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const fraction = numerator / denominator;
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const numString = numerator.toString();
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const denString = denominator.toString();
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if (numString[1] === '0' && denString[1] === '0') {
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// trivial
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return false;
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}
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for (let i = 0; i < 2; i++) {
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for (let j = 0; j < 2; j++) {
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if (numString[i] === denString[j]) {
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const newNum = parseInt(numString[1 - i], 10);
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const newDen = parseInt(denString[1 - j], 10);
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if (newNum / newDen === fraction) {
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return true;
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}
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}
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}
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}
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return false;
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}
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function findLargestDivisor(a, b) {
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let gcd = a > b ? b : a;
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while (gcd > 1) {
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if (a % gcd === 0 && b % gcd === 0) {
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return gcd;
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}
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gcd--;
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}
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return gcd;
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}
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function simplifyFraction(numerator, denominator) {
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const divisor = findLargestDivisor(numerator, denominator);
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return [numerator / divisor, denominator / divisor];
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}
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let multipleNumerator = 1;
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let multipleDenominator = 1;
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for (let denominator = 11; denominator < 100; denominator++) {
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for (let numerator = 10; numerator < denominator; numerator++) {
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if (isCurious(numerator, denominator)) {
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multipleNumerator *= numerator;
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multipleDenominator *= denominator;
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}
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}
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}
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return simplifyFraction(multipleNumerator, multipleDenominator)[1];
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}
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```
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