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.

Well, $d$ divides $a$, so in any $\Bbb Z_k$ it should be clear that $[a]\in\langle[d]\rangle$, whence $\langle[a]\rangle\subseteq\langle[d]\rangle$.
The reverse inclusion doesn't generally hold, but since $d=\gcd(a,n)$, then there exist $x,y\in\Bbb Z$ such that $d=ax+ny$, so in $\Bbb Z_n$ we have $[d]=[a][x]$, which lets us demonstrate the reverse inclusion in a similar fashion.
