Is $\mathbb{Z}_{10}^*$ cyclic or not? [duplicate]
Note that
$$\Bbb Z_{10}^*=\{[a]_{10}\in \Bbb Z_{10}\mid \exists [b]_{10}\in\Bbb Z_{10}, [a]_{10}\times_{10}[b]_{10}=[1]_{10}\},$$
where
$$[a]_{10}\times_{10}[b]_{10}=[ab]_{10}.$$
Its order is $\varphi(10)=4$, where $\varphi$ is Euler's totient function. (Why?)
Since
$$\begin{align} [3]_{10}^2&=[9]_{10}, \\ [3]_{10}^3&=[3]_{10}\times_{10}[9]_{10}= [7]_{10}, \end{align}$$
and $[3]_{10}^4=[3]_{10}\times_{10}[7]_{10}= [1]_{10}$, we must have that $\Bbb Z_{10}^*$ is generated by $[3]_{10}$.
Hence $\Bbb Z_{10}^*$ is cyclic.