Cyclic Group Elements whose Powers equal Identity

Theorem
Let $$G$$ be a cyclic group whose identity is $$e$$ and whose order is $$n$$.

Let $$d \backslash n$$.

Then there exist exactly $$d$$ elements $$x \in G$$ satisfying the equation $$x^d = e$$, and these are the elements of $$G_d = \left \langle {g^{n / d}} \right \rangle$$.

Proof
From the argument in Subgroup of Cyclic Group whose Order Divisor, it follows that $$x$$ satisfies the equation $$x^d = e$$ iff $$x$$ is a power of $$g^{n/d}$$.

Thus there are $$d$$ solutions to this equation.