TITLE: Find all integers $a,b,c$ such that $\frac {abc}{a+b+c}$ is an integer.
QUESTION [6 upvotes]: Find all integers $ (a,b,c) $ such that $ \frac{abc}{a+b+c} $ is an integer.
Could someone help me with this number theory problem?

REPLY [4 votes]: Note that $$\frac{abt}{a+b+t}=ab-\frac{ab(a+b)}{a+b+t}.$$
So pick any $a,b$ and list the divisors (positive and negative} of $ab(a+b)$ and then solve $a+b+t=d$ for each divisor $d.$