TITLE: Solve an equation in a cyclic group
QUESTION [5 upvotes]: Let $G$  be a finite cyclic group of order $n$. If $d$ is a  positive divisor of $n$, prove that equation $x^d = e$ has exactly $d$ distinct solutions in $G$
This is practice question assigned to me and I have no clue how to approach this.

REPLY [3 votes]: If you are better at number theory, you can start with this.
Let $g\in G$ be a generator and $f:G\rightarrow Z_n$ be defined by $f(g^i)=i$. You can easily show that $f$ is an isomorphism between $G$ and $Z_n$ which are just integers modulo $n$. Then, your problem reduces to $dx =0 (\text{mod} \ n)$.