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

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?

• I guess you mean solutions for integer $x$ and $y$? – Simon Feb 25 '12 at 10:10
• yes..obviously. – Amol Sharma Feb 25 '12 at 10:39
• Not obviously! I had to make that assumption. – fretty Feb 25 '12 at 10:44
• @Amol: Your question was not tagged diophantine-equations and your English is not perfect (not that I hold that against you), so it was not completely obvious. Maybe you meant Gaussian integers... Anyway, click on the question mark in my first comment. – Simon Feb 25 '12 at 10:57
• @amol Sharma: The answer depends on whether you mean integer solutions or positive integer solutions. If it is integers, there are $98$, if positive integers it is $49$. – André Nicolas Feb 25 '12 at 16:52

$1000(x+y) = xy$
$(x - 1000)(y - 1000) = 1000^2$
So that your solutions for $x-1000$ and $y-1000$ correspond to divisors of $1000^2$.
• ... and there are $49$ divisors of $10^6$ – Henry Feb 25 '12 at 13:48