Let $a, b, c$ be integers. Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
First time asking here. I'm not sure what your policies are on general homework help but I truly am stuck.
So far I have shown $\gcd(a,c) \gcd(b,c)$ as an integer combination of $ab$ and $c$. So if I can show that $\gcd(a,c) \gcd(b,c)$ divides $ab$ and $c$ I can use the proof that if an integer $d$ is a common divisor of $a$ and $b$, and $d=ax+by$ for some $x$ and $y$, that $d=\gcd(a,b)$. However I don't really know where to start with this. Any help would be appreciated.