---
id: 5900f38d1000cf542c50fea0
title: 'Problem 33: Digit cancelling fractions'
challengeType: 5
forumTopicId: 301987
dashedName: problem-33-digit-cancelling-fractions
---

# --description--

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.

We shall consider fractions like, <sup>30</sup>/<sub>50</sub> = <sup>3</sup>/<sub>5</sub>, to be trivial examples.

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.

If the product of these four fractions is given in its lowest common terms, find the value of the denominator.

# --hints--

`digitCancellingFractions()` should return a number.

```js
assert(typeof digitCancellingFractions() === 'number');
```

`digitCancellingFractions()` should return 100.

```js
assert.strictEqual(digitCancellingFractions(), 100);
```

# --seed--

## --seed-contents--

```js
function digitCancellingFractions() {

  return true;
}

digitCancellingFractions();
```

# --solutions--

```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];
}
```