Fix a positive integer $d$ and a positive divisor $k$ of $\varphi(d)$, and suppose there exists a primitive root modulo $d$. Use the result of part (b) to prove that there are precisely $\varphi(k)$ elements of $(\mathbb{Z}/d\mathbb{Z})^\times$ whose order is $k$. By taking $k=\varphi(d)$, deduce that there are $\varphi(\varphi(d))$ primitive roots modulo $d$.
In part(b), I proved that there exists $\ell\in\mathbb{Z}$ for which $\overline{c}=\overline{b}^\ell$, and $\overline{c}\in\left\{\overline{1},\overline{b},\ldots,\overline{b}^{k-1}\right\}$. I'm struggling to find out how to use them to prove this statement...
