# 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 $4527$:

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

$\blacksquare$

## 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.