Let n be be positive integer such that $n=\frac{pb}{q}$ for some prime number $p$ and for some integers $b$ and $q$. Let's also assume that $gcd(p,q)=1$. Show that $p|n$.
My attempt:
Since $n$ is positive integer we have that $q|pb$ and since $p$ and $q$ are relativity prime we have that $q|b$ so $\frac{b}{q}=a$, where $a$ is some integer. So we have that $n=ap$ and this ends the proof.
Is my reasoning correct?
