---
id: 5a23c84252665b21eecc7eca
title: Kaprekar numbers
challengeType: 5
---

## Description
<section id='description'>
A positive integer is a <a href="https://en.wikipedia.org/wiki/Kaprekar number">Kaprekar number</a> if:
<ul>
<li>It is 1, or,</li>
<li>The decimal representation of its square may be split once into two parts consisting of positive integers which sum to the original number. </li>
</ul>
Note that a split resulting in a part consisting purely of 0s is not valid, as 0 is not considered positive.Example
Kaprekar numbers:
<ul>
<li> <code>2223</code> is a Kaprekar number, as <code>2223 * 2223 = 4941729</code>, <code>4941729</code> may be split to <code>494</code> and <code>1729</code>, and <code>494 + 1729 = 2223</code></li>
<li>The series of Kaprekar numbers is known as <a href="http://rosettacode.org/wiki/oeis:A006886">A006886</a>, and begins as <code>1, 9, 45, 55, ...</code></li>
</ul>
</section>

## Instructions
<section id='instructions'>
Write a function that takes a number $n$, a base $bs$, and returns true if the number is a Kaprekar number for the given base. Otherwise, the function returns false.
</section>

## Tests
<section id='tests'>

``` yml
tests:
  - text: <code>isKaprekar</code> should be a function.
    testString: assert(typeof isKaprekar == 'function', '<code>isKaprekar</code> should be a function.');
  - text: <code>isKaprekar(1, 10)</code> should return a boolean.
    testString: assert(typeof isKaprekar(1, 10) == 'boolean', '<code>isKaprekar(1, 10)</code> should return a boolean.');
  - text: <code>isKaprekar(1, 10)</code> should return <code>true</code>.
    testString: assert.equal(isKaprekar(1, 10), true, '<code>isKaprekar(1, 10)</code> should return <code>true</code>.');
  - text: <code>isKaprekar(9, 10)</code> should return <code>true</code>.
    testString: assert.equal(isKaprekar(9, 10), true, '<code>isKaprekar(9, 10)</code> should return <code>true</code>.');
  - text: <code>isKaprekar(2223, 10)</code> should return <code>true</code>.
    testString: assert.equal(isKaprekar(2223, 10), true, '<code>isKaprekar(2223, 10)</code> should return <code>true</code>.');
  - text: <code>isKaprekar(22823, 10)</code> should return <code>false</code>.
    testString: assert.equal(isKaprekar(22823, 10), false, '<code>isKaprekar(22823, 10)</code> should return <code>false</code>.');
  - text: <code>isKaprekar(9, 17)</code> should return <code>false</code>.
    testString: assert.equal(isKaprekar(9, 17), false, '<code>isKaprekar(9, 17)</code> should return <code>false</code>.');
  - text: <code>isKaprekar(225, 17)</code> should return <code>true</code>.
    testString: assert.equal(isKaprekar(225, 17), true, '<code>isKaprekar(225, 17)</code> should return <code>true</code>.');
  - text: <code>isKaprekar(999, 17)</code> should return <code>false</code>.
    testString: assert.equal(isKaprekar(999, 17), false, '<code>isKaprekar(999, 17)</code> should return <code>false</code>.');
```

</section>

## Challenge Seed
<section id='challengeSeed'>
  <div id='js-seed'>

```js
function isKaprekar(n, bs) {
  // Good luck!
}
```

  </div>
</section>

## Solution
<section id='solution'>

```js
function isKaprekar(n, bs) {
  if (n < 1) return false;
  if (n == 1) return true;
  for (var a = n * n, b = 0, s = 1; a; s *= bs) {
    b += a % bs * s;
    a = Math.floor(a / bs);
    if (b && a + b == n) return true;
  } return false;
} 
```

</section>