TITLE: Solve $\frac1x+\frac1y=\frac1{pq}$
QUESTION [7 upvotes]: For $x,y\in\mathbb{N}$ how many ordered pairs $\left(x,y\right)$ satisfy   $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{pq}$ where $p,q$ are distinct primes?

REPLY [8 votes]: The "easy way" to deal with $\frac 1x+\frac 1y=\frac 1N$ is to set $x=N+a, y=N+b$ and clear fractions to obtain $ab=N^2=p^2q^2$
Now $p^2q^2$ has $(2+1)(2+1)$ factors so there should be nine solutions for $a$ - from which $b$ is determined.