Which of the following is isomorphic to the group of units of $\mathbb{Z}_{35}$?

Question

• $C_2 \times C_2 \times C_6$
• $C_2 \times C_{12}$
• $C_{24}$

The group of units of $\mathbb{Z}_{35}$ is $\mathbb{Z}_{35}^\#= \left\{ 1,2,3,4,6,8,9,11,12,13,16,17,18,19,22,23,24,26,27,29,31,32,33,34 \right\}$.

Answer 1

Hint: $\mathbb{Z}_{35} \cong \mathbb{Z}_{5} \times \mathbb{Z}_{7}$, by the Chinese remainder theorem.

Answer 2

Chinese Remainder Theorem, so $C_4 \times C_6$. Then as pointed out below, you can write as $C_2 \times C_3 \times C_4 $ and thence as $C_2 \times C_{12}$, clumsy though it is.