Let $M=\mathbb{Z}^4/N$ where $N$ is a subgroup of $\mathbb{Z}^4$ generated by $(1,0,-1,3)$ and $(2,4,8,-6)$. Recognize $M$ as a product of cyclic groups.
Here I have to use the following Theorem:
If $R$ is a PID and $M$ is a finitely generated $R-$ module, then $M\cong R\oplus R\oplus...\oplus R\oplus R/(a_1)\oplus ... \oplus R/(a_m)$ such that $a_1|a_2|...|a_{m-1}|a_m$.
Can anyone please help me how to apply this theorem to solve the problem?