Proving the Existence of Subgroups in Cyclic Groups

[screenname123]

Homework Statement

Let G be a finite cyclic group of order n. If d is a positive divisor of n, prove that the equation x^d=e has d distinct solutions

Homework Equations

n=dk for some k
order(G)=n

The Attempt at a Solution

solved it:
<g^k>={g^k, g^2k,...,g^dk=e} and for all x in <g^k> x^d=e and order(g^k)=d.

 

[jbunniii]

What can we presume that you already know about cyclic groups? Do you know the theorem that if $d$ divides $n$, then $G$ has a subgroup of order $d$? If not, then I would start by proving that. Your result will follow immediately from that theorem.