Express a group in four different direct product

Question

Question: Express U (165) as an internal dirext product of proper subgroup in four different ways.

The hint suggests the use of the Chinese remainder theorem which I am unfamiliar with. Is there another way around this theorem?

Thanks in advance

Answer 1

Hint: The Chinese Remainder Theorem says that for any ring $R$ and two ideals $I,J\subseteq R$ with $I+J=R$ that we have a natural iso $R/IJ\cong R/I\times R/J$. Apply this theorem to the ring $\mathbb{Z}$ and the ideals $n\mathbb{Z}$ and $m\mathbb{Z}$ for $m,n$ coprime.