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---
id: 5900f38d1000cf542c50fea0
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
assert(typeof digitCancellingFractions() === 'number');
```
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`digitCancellingFractions()` should return 100.
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```js
assert.strictEqual(digitCancellingFractions(), 100);
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```
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# --seed--
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## --seed-contents--
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```js
function digitCancellingFractions() {
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return true;
}
digitCancellingFractions();
```
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# --solutions--
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```js
function digitCancellingFractions() {
function isCurious(numerator, denominator) {
const fraction = numerator / denominator;
const numString = numerator.toString();
const denString = denominator.toString();
if (numString[1] === '0' & & denString[1] === '0') {
// trivial
return false;
}
for (let i = 0; i < 2 ; i + + ) {
for (let j = 0; j < 2 ; j + + ) {
if (numString[i] === denString[j]) {
const newNum = parseInt(numString[1 - i], 10);
const newDen = parseInt(denString[1 - j], 10);
if (newNum / newDen === fraction) {
return true;
}
}
}
}
return false;
}
function findLargestDivisor(a, b) {
let gcd = a > b ? b : a;
while (gcd > 1) {
if (a % gcd === 0 & & b % gcd === 0) {
return gcd;
}
gcd--;
}
return gcd;
}
function simplifyFraction(numerator, denominator) {
const divisor = findLargestDivisor(numerator, denominator);
return [numerator / divisor, denominator / divisor];
}
let multipleNumerator = 1;
let multipleDenominator = 1;
for (let denominator = 11; denominator < 100 ; denominator + + ) {
for (let numerator = 10; numerator < denominator ; numerator + + ) {
if (isCurious(numerator, denominator)) {
multipleNumerator *= numerator;
multipleDenominator *= denominator;
}
}
}
return simplifyFraction(multipleNumerator, multipleDenominator)[1];
}
```