Factor group of a finite cyclic group

Question

Prove that $$\mathbb Z_m/\langle \overline{n}\rangle \cong\mathbb Z_{\text{gcd}(m,n)}.$$ for any $m,n\in \mathbb{N}$.

For this, I know I must show that in $\mathbb Z_m$, $\langle \overline{n} \rangle$ and $\langle \text{gcd}(m,n)\rangle$ are the same ideal, but I'm not sure how I could possibly show that. Any suggestions?

Answer 1

Hint: Consider the projection $\Bbb Z_m \to \Bbb Z_{gcd(m,n)}$ and the first isomorphism theorem.