Prove that if gcd(a,b)=1 then $\forall x \in \mathbb{Z}$ ab div x $\iff$ a div x $\land$ b div x

Question

I've managed to prove the direction ab div x $\implies$ a div x $\land$ b div x, but I'm struggling to see how I should use the fact that gcd(a,b)=1 for the other direction (a div x $\land$ b div x $\implies$ ab div x) of the proof.

I start by assuming a div x $\land$ b div x which allows us to say $x=a\cdot h=b\cdot m$ where $h,m\in\mathbb{Z}$, but I'm failing to see how to use gcd(a,b)=1 to move further along the proof.