Given a rational number $p/q$, show that the equation $\frac{1}{x} + \frac{1}{y} = \frac{p}{q}$ has only finite many positive integer solutions. How can i solve this,
Given a rational number $p/q$, show that the equation $\frac{1}{x} + \frac{1}{y} = \frac{p}{q}$ has only finite  many positive integer solutions.
I thought let $\frac pq=\frac1r$, but solving this gets $(x-r)(y-r)=r^2$, which gives the no. of solutions to be $k(r^2)$ where $k(r^2)$ is the no. of divisors of $r$.
Thanks.

Hint: A solution always satisfies
$$ \frac{x + y}{xy} = \frac{p}{q} $$.
Conclude that $xy = r q$ for some positive integer $r$. Now what?
