Let $p$ be an odd prime number. Assuming that the multiplicative group of $\mathbb{Z} / p \mathbb{Z}$ is cyclic, prove that the multiplicative group of units of $\mathbb{Z} / p^{n} \mathbb{Z}$ is cyclic for all $n \geqslant 1$.

Find an integer $a$ such that its residue class in $\mathbb{Z} / 11^{n} \mathbb{Z}$ generates the multiplicative group of units for all $n \geqslant 1$.