Prove that $\gcd(k,mn) = 1$ iff $\gcd(k,n) = 1$ and $\gcd(k,m) = 1$

Question

Reverse direction I have:

By $k,n$ being coprime, there are integers $x,y$ such that $xk + ny=1$. Likewise, we can write $ak+bm=1$. Hence, we have:

$$xk+ny = ak+bm = 1.$$

But from here, I am not sure how to proceed. Likewise, with forward direction. Just need a hint to continue.