Integer is Congruent Modulo Divisor to Remainder

Theorem
Let $a \in \Z$.

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

Then:
 * $a \equiv r \pmod m$

where the notation denotes that $a$ and $r$ are congruent modulo $m$.

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

Then:
 * $\exists q \in \Z: a = q m + r$

Hence by definition of congruence modulo $m$:
 * $a \equiv r \pmod m$