In $\mathbb Z_n$ prove that $\langle h,k\rangle = \langle\gcd(h,k,n)\rangle$

Question

In $\mathbb Z_n$ prove that $$\langle h,k\rangle = \langle\gcd(h,k,n)\rangle$$ where $h,k \in \mathbb Z_n$.

I can prove that in $\mathbb Z$,$$\langle h,k\rangle = \langle\gcd(h,k)\rangle$$ using Bezout's Identity but I need hints for the $\mathbb Z_n$ case.

Relevant: 1,2

Answer 1

Hint: For any ring $R$ and any ideals $I, J\subset R$, one has $$I\cdot R/J=(I+J)/J,$$ so $\;\langle h,k\rangle\cdot \mathbf Z/n\mathbf Z=\langle h,k,n\rangle/n\mathbf Z $.