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:


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

That is, GCD is multiplicative.