Number of integer solutions of $\frac{1}{x} + \frac{1}{y} = \frac{1}{1000}$

Question

What is the number of integer solutions of: $$\frac{1}{x} + \frac{1}{y} = \frac{1}{1000}$$ How to solve these type of problems if am comfortable of solving $x+y=z$. But how to do if multiplicative inverses are involved?

Answer 1

Assuming you mean integer solutions, you will be able to rewrite your equation as:

$1000(x+y) = xy$

Then rearranging you will be able to write as:

$(x - 1000)(y - 1000) = 1000^2$

So that your solutions for $x-1000$ and $y-1000$ correspond to divisors of $1000^2$.