Integer is Congruent Modulo Divisor to Remainder

Theorem
If $$a \in \Z$$ has a principal 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 principal remainder when divided by $$m$$.

Proof
Let $$a$$ have a principal 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 Modulo m Equivalence.