Congruence as an equivalence relation

Definition. $a\equiv b\pmod n\iff n\mid a-b\iff\exists k\in\mathbb{Z}:a=b+kn$.

Reflexive. $a-a=0=n\cdot0$.

Symmetric. If $a-b=kn$, then $b-a=(-k)n$.

Transitive. If $a-b=kn$ and $b-c=\ell n$, then $a-c=(k+\ell)n$.

Conclusion. Congruence is an equivalence relation whose classes are the residue classes mod $n$.