Find all positive integer solutions $(a, b)$ to the equation $$\frac{1}{a}+\frac{1}{b}+ \frac{n}{lcm(a,b)}=\frac{1}{gcd(a, b)}$$for (i) $n = 2007$; (ii) $n = 2010$.
Problem
Source: New Zealand NZMOC Camp Selection Problems 2010 Juniors 4
Tags: LCM, GCD, least common multiple, greatest common divisor, number theory