A number theory problem by Sachin Vishwakarma

Number Theory Level 4

Let $p$ and $q$ be distinct primes. Find the number of positive integer solutions of the equation $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{pq}$ .

1 solution

Aditya Narayan Sharma
Feb 24, 2016

$\frac{(x + y)}{xy}$ = $\frac{1}{pq}$

or, (x + y)pq = xy

or, xy - (x + y)pq + $p^{2}q^{2}$ = $p^{2}q^{2}$

or, (x - pq)(y - pq) = $p^{2}q^{2}$

So number of solutions = (2+1)(2+1) = 9