Suppose that a is an element of $\mathbb{Z}/_{101}\mathbb{Z}$ such that $a^{513} = 1$. Is it possible that $a$ is a primitive $100$-th root of $1$ in $\mathbb{Z}/_{101}\mathbb{Z}$? Either give an example of a primitive element whose $513$rd power is $1$ or prove that there does not exist such an element.
Ok so I understand that if $a$ is a primitive $100$th root of $1$ in $\mathbb{Z}/_{101}\mathbb{Z}$, then the smallest exponent of $a$ which equals $1$ mod$(101)$ is $100$. This means that if $a$ is a primitive root, then
$a^{100}=1$ mod$(101)$
So how could I show that? What can I do from here?