GCD from Congruence Modulo m

Theorem
Let $a, b \in \Z, m \in \N$. Let $a$ be congruent to $b$ modulo $m$.

Then the GCD of $a$ and $m$ is equal to the GCD of $b$ and $m$.

That is:
 * $a \equiv b \pmod m \implies \gcd \set {a, m} = \gcd \set {b, m}$

Proof
We have:
 * $a \equiv b \pmod m \implies \exists k \in \Z: a = b + k m$

Thus:
 * $a = b + k m$

and the result follows directly from GCD with Remainder.