elementary number theory - If $gcd(ab,c)=d$ and $c|ab$ then $c=d$

For all positive integers $a$, $b$, $c$ and $d$,

if $\gcd(ab, c) = d$ and $c | ab$, then $c = d$.

Need help proving this question, I know that $abx + cy = d$ for integers $x,y$
and that $c|ab$ can be $c=q\cdot ab$ but I'm not sure how to apply these facts or if they're even useful in this proof.

Any help to get me started would be great.

Answer

$c|ab$ means that $ab=q\cdot c$, not the other way around!

Therefore, you have $ab=q\cdot c$ and $ab x + c y = d$ which you can rewrite into $$qcx + cy = d\\ c(qx+y) = d$$

Can you continue from here?