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.