elementary number theory - Prove the following: If $a mid bc$, then $a mid gcd(a, b)c$.

Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.

I tried to set $\gcd(a, b)$ to $b$ and used the fundamental theorem of arithmetic to prove that it is divisible by $a$, but I can't prove that $a \mid bc$, if and only if $a\mid b$ and $a\mid c$. Please help. Thanks.