GCD with One Fixed Argument is Multiplicative Function

Theorem
Let:
 * $a, b, c \in \Z: b \perp c$

where $b \perp c$ denotes that $b$ is coprime to $c$.

Then:
 * $\gcd \set {a, b} \gcd \set {a, c} = \gcd \set {a, b c}$

That is, GCD is multiplicative.