Definition:Residue Class

Let $$m \in \Z, m > 0$$.

Let $$\mathcal{R}_m$$ be the congruence relation modulo $m$on the set of all $$a, b \in \Z$$:


 * $$\mathcal{R}_m = \left\{{\left({a, b}\right) \in \Z \times \Z: \exists k \in \Z: a = b + km}\right\}$$

We have that congruence modulo $m$ is an equivalence relation.

So for any $$m \in \N$$, we denote the equivalence class of any $$a \in \Z$$ by $$\left[\!\left[{a}\right]\!\right]_m$$, such that:

$$ $$

The equivalence class $$\left[\!\left[{a}\right]\!\right]_m$$ is called the congruence class of $$a$$ (modulo $$m$$).

It follows directly from the definition of equivalence class that $$\left[\!\left[{x}\right]\!\right]_m = \left[\!\left[{y}\right]\!\right]_m \iff x \equiv y \left({\bmod\, m}\right)$$.

These congruence classes are known as residue classes.