# Definition:Kaprekar's Process

## Definition

Kaprekar's process is the repeated application of the Kaprekar mapping to a given positive integer.

## Examples

### Kaprekar's Process on $4527$

From $4527$:

 $\displaystyle 7542 - 2457$ $=$ $\displaystyle 5085$ $\displaystyle 8550 - 0558$ $=$ $\displaystyle 7992$ $\displaystyle 9972 - 2799$ $=$ $\displaystyle 7173$ $\displaystyle 7731 - 1377$ $=$ $\displaystyle 6354$ $\displaystyle 6543 - 3456$ $=$ $\displaystyle 3087$ $\displaystyle 8730 - 0378$ $=$ $\displaystyle 8352$ $\displaystyle 8532 - 2358$ $=$ $\displaystyle 6174$ $\displaystyle 7641 - 1467$ $=$ $\displaystyle 6174$

$\blacksquare$

## Also known as

Kaprekar's process is also known as the Kaprekar routine or the Kaprekar sequence.

Sometimes rendered as the Kaprekar process.

## Also defined as

Some sources define the Kaprekar mapping so as not to retain the leading zeroes, and so, for example:

$K \left({1121}\right) = 2111 - 1112 = 999$
$K \left({999}\right) = 999 - 999 = 0$

$K \left({1121}\right) = 2111 - 1112 = 0999$
$K \left({0999}\right) = 9990 - 0999 = 8991$

The process as initially specified does retain all leading zeroes.

## Source of Name

This entry was named for Dattathreya Ramchandra Kaprekar.