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.

WLOG you may assume $\frac{1}{x}$ $\geq$ $\frac{1}{y}$. 
Then $\frac{1}{x}\geq\frac{1}{2}.\frac{p}{q}$ so $\frac{2p}{q}\geq x$.
There are only finite many positive integers no larger than $2p/q$, also for each $x$ there is only one $y$ ,therefore there are finite solutions.
