[Math] If $x$ divides $z$, $y$ divides $z$ and $x$ and $y$ are relatively prime, then $xy$ divides $z$.

Question

I've been stuck on this for far too long, but it isn't proved in my lecture notes so it's supposed to be elementary…

Let $x, y, z$ be integers such that $x$ divides $z$, $y$ divides $z$ and $x$ and $y$ are relatively prime, then $xy$ divides $z$.

Thanks a lot !

Answer 1

$\gcd(x,y)=1$ $\implies$ there are integers $s,t$ such that $sx+ty=1$.

Let $z=ax=by$. Then $sx+ty=1$ $\implies$ $sxz+tyz=z$, i.e. $(sb+ta)xy=z$. Thus $xy\mid z$.