The number of solutions of $x^n = e$ in a finite group is a multiple of n, whenever n divides the group order.

Question

Prove that in a finite group G the number of solutions of the equation $x^n = e$ is a multiple of n, whenever n divides the order of the group.

I feel there is a very simple answer to this question, but it eludes me.

Answer 1

Hint: The number of generators of a cyclic group of order $d$ is $\phi(d)$, where $\phi$ is Euler's totient function. Thus the number of elements of order $d$ in a group must be a multiple of $\phi(d)$.

What you are looking for is the sum of the numbers of elements of order $d$, where $d \mid n$.