Why can $\frac{a}{(a,b)}|(\frac{b}{(a,b)})*(x-y)$ be simplified as $\frac{a}{(a,b)}|(x-y)$?

Question

I am reading the proof for a proposition in a number theory book, and it states that by another proposition which states $(\frac{a}{(a,b)}, \frac{b}{(a,b)})=1$, where $(a,b)$ is the greatest common divisor of $a$ and $b$, $\frac{a}{(a,b)}|(\frac{b}{(a,b)})*(x-y)$ be simplified as $\frac{a}{(a,b)}|(x-y)$. How does that simplification occur?

Answer 1

Because $\frac{a}{(a,b)}$ and $\frac{b}{(a,b)}$ are relatively prime.