# Congruence Modulo Integer/Examples/12321 equiv 111 mod 3

## Example of Congruence Modulo an Integer

$12 \, 321 \equiv 111 \pmod 3$

## Proof

By definition of congruence:

$x \equiv y \pmod n$ if and only if $x - y = k n$

for some $k \in \Z$.

We have:

$12 \, 321 - 111 = 12 \, 210$

The digit sum of $12 \, 210$ is $3$.

By Divisibility by 3, it follows that:

$12 \, 210 = k \times 3$

for some $k \in \Z$.

The result follows by definition of congruence.

$\blacksquare$