Prove that if $\gcd(a, n) = d$ then $\langle[a]\rangle = \langle[d]\rangle$ in $\mathbb Z_n$? I am not sure how to start this problem and hope someone can help me out.

By Bezout, there exist $u,v$ such that $d=ua+vn$. It follows that $\langle[d]\rangle\subseteq \langle[a]\rangle$. 
And since $d|a$, $[a]=k[d]$ belongs to $\langle[d]\rangle$ so the reverse inclusion holds.
