Proving Least Common Multiple

Question

So I completed the first part of the proof where I prove $\frac{xy}{gcd(x,y)}$ is a multiple of $x$ and $y$. Now I need to prove for the second part that for any integer $m$ such that $x | m$ and $y | m$, $xy | (m*gcd(x,y))$ and then use these parts to prove the final part that $lcm(x,y) = \frac{xy}{gcd(x,y)}$. I'm confused on how to approach the second part and I also don't see how to connect the two parts when I do. Any help is appreciated

Answer 1

Let $k=\gcd(x,y)$ and write $x=ak, y=bk$. You need to show $abk|m$ so that $abk^2|mk$.

To do this you write $m=pak=qbk$ and simplify to $pa=qb$. Then use $\gcd(a,b)=1$ to infer $b|p$. Therefore $p=rb$ and $m=rabk$.