For each positive integer $n$ prove that $\textbf{C}^{*}$ the group of nonzero complex numbers under multiplication

Question

For each positive integer $n$ prove that $\textbf{C}^{*}$ the group of nonzero complex numbers under multiplication , has exactly $\phi(n)$ elements of order $n$. Where $\phi$ is totient function.

I was thinking about this problem and my answer was

let $H=\{x \in \textbf{C}|x^{n}=1\}$ then for each positive integer $n$, this group is cyclic. Since there are $\phi(n)$ generators in each of these cyclic groups for each $n$, there are exactly $\phi(n)$ elements of order $n$ in $\textbf{C*}$.