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:


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

That is, GCD is multiplicative.