The multiplicative group of $\mathbb{Z}/2^n \mathbb{Z}$ is $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2^{n-2}\mathbb{Z}$ and is non-cyclic whenever $n \ge 3$. The multiplicative group of $\mathbb{Z}/3^n \mathbb{Z}$ is cyclic of order $\phi(3^n) = 2 \times 3^{n-1}$.