Invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$

Question

Find the invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$.

I have shown that $(\mathbb{Z}/77 \mathbb{Z})^{\times} \cong \mathbb{Z}/10 \mathbb{Z} \times \mathbb{Z}/6 \mathbb{Z}$. I am stuck on this problem. Could anyone help me at this point?

Answer 1

Hint: By the Chinese remainder theorem, $$ C_{10} \cong C_2 \times C_5, \qquad C_{6} \cong C_2 \times C_3 $$ Rearrange and recombine. This gives the elementary divisors $$ C_2 \times C_2 \times C_3 \times C_5 $$ and the invariant factors $$ C_2 \times (C_2 \times C_3 \times C_5) \cong C_2 \times C_{30} $$ and