Definition:Congruence Class Modulo m

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

If $$r$$ is the smallest non-negative integer in $$\left[\left[{a}\right]\right]_m$$, then $$0 \le r < m$$ and $$a \equiv r \left({\bmod\, m}\right)$$ from Congruence to an Integer less than Modulus.

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 $$\left[\left[{x}\right]\right]_m = \left[\left[{y}\right]\right]_m \iff x \equiv y \left({\bmod\, m}\right)$$.