elementary number theory - Prove if $nmid ab$, then $nmid [gcd(a,n) times gcd(b,n)]$

Prove if $n\mid ab$, then $n\mid [\gcd(a,n)\times \gcd(b,n)]$

So I started by letting $d=\gcd(a,n)$ and $e=\gcd(b,n)$.
Then we have $x,y,w,z$ so that $dx=a$, $ey=b$,$dw=ez=n$
and we also have $s$ so that $ns=ab$

or $ns=dexy$.

what I want is $n\mid de$, but I'm only getting to $n\mid de(xy)$ since I cannot prove that $s/(xy)$ is an integer.