Under what conditions on $p$ and $q$, the integers $a$ and $b$ have a common prime divisor

Question

Let us consider a formula of the form:

$$a=(p/q)b$$ where $a, p,q,b$ are positive integers such that $p$ and $q$ are coprime.

My question is: Under what conditions on $p$ and $q$ do the integers $a$ and $b$ have a common prime divisor?

Answer 1

There is no connection between the values of $p$ and $q$ and the common factors of $a$ and $b$. This is because for any given coprime $p,q$ we can choose any number $r$ and set $a=pr$, $b=qr$. This gives a solution where $\gcd(a,b)=r$, so for any $p,q$ we can get any gcd we want.