Congruence

Two integers $a$ and $b$ are congruent modulo $n$ (written $a \equiv b \pmod{n}$) if $n$ divides $a - b$.

Congruence modulo $n$ is an equivalence relation on $\mathbb{Z}$. It is compatible with addition and multiplication: if $a \equiv a' \pmod{n}$ and $b \equiv b' \pmod{n}$, then $a + b \equiv a' + b' \pmod{n}$ and $ab \equiv a'b' \pmod{n}$. This compatibility is what makes the quotient group $\mathbb{Z}/n\mathbb{Z}$ a ring.