Congruence class ($[a{]}_N$)

The congruence class of $a$ modulo a positive integer $N$ is $\{a+Nk:k\in \mathbb{Z}\}$. Two integers have the same class exactly when their difference is divisible by $N$. Addition and multiplication of representatives descend to well-defined operations on $\mathbb{Z}/N\mathbb{Z}$, because replacing a representative by a multiple-of-$N$ translate changes the result by a multiple of $N$.

A complete residue system modulo a positive integer $m$ contains exactly one representative of every congruence class modulo $m$. Both $\{0,\dots ,m-1\}$ and $\{1,\dots ,m\}$ are examples. Counting the representatives that are coprime to $m$ gives Euler's totient function; those representatives form a reduced residue system.