Consider the set of $d$-tuples
\begin{eqnarray} S=\{(g_1, \cdots, g_d)|g_1g_2\cdots g_d=e, g_1, g_2, \cdots, g_d\text{ are not periodic with periodicity}>1\text{ and }<d\}\end{eqnarray}
Note that $|S|$ is divisible by $d$. Consider the $\mathbb{Z}_d$-action on $S$ by cyclically permuting the entries in a tuple, i.e. the generator $1\in\mathbb{Z}_d$ takes $(g_1, g_2, \cdots, g_d)$ to $(g_2, g_3, \cdots, g_d, g_1)$. There are two types of orbits, namely those tuples in $E_d$ and the orbits consisting of tuples with some entries distinct. Each tuple in $E_d$ forms an orbit itself, while each orbit of the other type consists of $d$ tuples. By class equation,
\begin{eqnarray}|S|=|E_d|+\text{#tuples with some entries distinct}\end{eqnarray}
Note that the set of tuples with some entries distinct is a disjoint union of orbits each of which consists of $d$ tuples. Thus $|E_d|$ is divisible by $d$.
Remark: This is just an adaption of the proof of Cauchy's theorem.