Integer is Congruent Modulo Divisor to Remainder

Theorem
If $a \in \Z$ has a remainder $r$ on division by $m$, then $a \equiv r \pmod m$.

Corollary
$a \equiv b \pmod m$ iff $a$ and $b$ have the same remainder when divided by $m$.

Proof
Let $a$ have a remainder $r$ on division by $m$.

Then $\exists q \in \Z: a = qm + r$.

Hence $a \equiv r \pmod m$.

Proof of Corollary
Follows directly from the above and Congruence (Number Theory) is Equivalence Relation.