Elementary number theory proof involving multiplied gcd's

I'm having trouble proving the following if and only if statement:

For all integers $a,b,n$, prove that $n|gcd(a,n)gcd(b,n)$ if and only if $n|ab$

For proving $n|gcd(a,n)gcd(b,n)\implies n|ab$, I tried using Bezout's Lemma for both $gcd$s and expanding but didn't know how to show that $n$ divided $ab$.