Proof with roots of unity

Question

Let $m,n \in \mathbb N$ and $d=gcd(m,n).$
Prove that if w is both an m-th root of unity and an n-th root of unity, then w is a d-th root of unity.

How would i begin about starting this type of proof?

Answer 1

Hint $ $ The set $E\,$ of exponents $\,k\,$ such that $\,w^k = 1\,$ are closed under subtraction so, iterating, also closed under mod, so also under gcd. $\ $ Or, explicitly, use the Bezout identity for the gcd.