I came across this problem in my number theory text and am having a bit of trouble with it:

Prove if $c\mid ab$ and $\gcd(c,a)=d$, then $c\mid db$.

Here's what I have so far:

If $c\mid ab$, then there exists an integer $x$ such that $cx=ab$.  

Because $\gcd(c,a)=d$, $d\mid c$ and $d\mid a$. Let $y$ be such that $dy=a$.   

Then,
$$cx=ab=dyb,$$
so $$cx \frac{d}{a} = db.$$
But I don't know how to show that $\frac{dx}{a}$ is an integer. Can anyone please offer ideas on what I've done so far?