Let $a, b, c \in\mathbb{Z}$ and assume $\mathrm{gcf} (a, b) = 1$ and $\mathrm{gcf} (a, c) =1$. Proof that gcf (a, bc) = 1.
$\gcd(a,b)\!=\!1\!=\!\gcd(a,c)\Rightarrow\gcd(a,bc)\!=\!1$ [coprimes to $\,a\,$ are product closed]