Let $a, b, c$ be nonzero integers such that $M = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $N =\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are both integers. Find $M$ and $N$.
Source: Indian Postal Coaching 2007 set 6 p2
Tags: Integers, number theory, algebra
Let $a, b, c$ be nonzero integers such that $M = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $N =\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are both integers. Find $M$ and $N$.