Find quantity of elements in group with given order

Question

Let $G = ( \mathbb { Z } / 133 \mathbb { Z } ) ^ { \times }$ be the group of units of the ring $\mathbb { Z } / 133 \mathbb { Z }$ . Find the number of elements of $G$ of order $9 .$

133 cannot be divided by 9. So what is the solution to the problem? Or my consideration is too simple to realize the problem precisely?

Answer 1

The fact that $9\nmid133$ is irrelevant here, since the group $\mathbb{Z}_{133}^\times$ has $108$ elements. Since $9\mid108$, Lagrange's theorem is not an obstacle to the existence of elements of order $9$.

Answer 2

Hint: $ ( \mathbb { Z } / 133 \mathbb { Z } ) ^ { \times } \cong ( \mathbb { Z } / 7 \mathbb { Z } ) ^ { \times } \times ( \mathbb { Z } / 19 \mathbb { Z } ) ^ { \times } \cong C_6 \times C_{18} $