If $d=(m, n)$, with $m, n>0$, do $a, b$ integers exist such that $1=(\frac{m}{a}, \frac{n}{b})$ and $d=ab$?

Question

I've been doodling around with some numbers and I think this is true, however I have not been able to prove it yet. I'd like to know if you have a suggestion or a counterexample for this. Thanks!