Congruence by Product of Moduli

Theorem
Let $a, b, m \in \Z$.

Let $a \equiv b \pmod m$ denote that $a$ is congruent to $b$ modulo $m$.

Then $\forall n \in \Z, n \ne 0$:
 * $a \equiv b \pmod m \iff a n \equiv b n \pmod {m n}$

Proof
Let $n \in \Z: n \ne 0$.

Then:

Hence the result.

Note the invalidity of the third step when $n = 0$.