Let $p$ be an odd prime. Prove that the multiplicative groups $\left(\mathbb{Z} / p^{n} \mathbb{Z}\right)^{\times}$are cyclic for $n \geqslant 2$. [You may assume that the multiplicative group $(\mathbb{Z} / p \mathbb{Z})^{\times}$is cyclic.]

Find an integer which generates $\left(\mathbb{Z} / 7^{n} \mathbb{Z}\right)^{\times}$for all $n \geqslant 1$, justifying your answer.