# What makes two numbers congruent?

Two whole numbers $a$ and $b$ are said to be congruent modulo a third natural number $m$ if and only if $m$ divides the difference between $a$ and $b$. Equivalently the remainders when the numbers are divided by $m$ are equal.
The expression "$a$ is congruent (or equivalent) to $b$ modulo $m$" is written in symbols as follows:
$a\equiv b\mod m$
The usual representative for the equivalence class of a given number modulo $m$ is the natural number in the range $[0,m)$ although it is often useful to use $-1$ rather than $m-1$ because of the properties of addition and multiplication modulo $m$.
Note that two numbers are congruent only modulo a third number. For example
$7\equiv5\equiv1\mod2$
but
$7\equiv1\mod3$
whereas
$5\equiv2\mod3$
So it does not make sense to say simply "7 is congruent to 5" without adding the modulo.

What makes two numbers congruent?