Integer is Congruent Modulo Divisor to Remainder

Theorem
If $$a \in \mathbb{Z}$$ has a principal remainder $$r$$ on division by $$m$$, then $$a \equiv r \left({\bmod\, m}\right)$$.

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

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

Hence $$a \equiv r \left({\bmod\, m}\right)$$.