The coprimality of 2 integers that are divisors of 2 larger coprimes?

Question

Given an example with $(a,b)=1$ where $a=ux$ and $b=vy$ (with all variables being integers), obviously $(ux,vy) = 1$ directly, but does $(u,v) = 1$ as well? I am pretty sure it should but I am unsure if this is actually a true statement and how a rigorous proof of this would look.

Answer 1

If $u$ and $v$ share a prime factor then so do $ux$ and $vy$. So if the latter two are coprime, so are the former.