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:


 * $\gcd \left\{ {a, b}\right\} \gcd \left\{ {a, c}\right\} = \gcd \left\{ {a, b c}\right\}$

That is, GCD is multiplicative.