Help me to find $\frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{<(1,1,1),(1,3,2)>}$. I have previously asked this question. But now I'm stuck in finding $\frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{<(1,1,1),(1,3,2)>}$.
Please give me hints to find it.

Since the matrix
$$
\begin{pmatrix}
1 & 1 & 1\\
1 & 3 & 0\\
1 & 2 & 1
\end{pmatrix}
$$
has determinant $1$, then $\{v_1, v_2, v_3\} = \{(1,1,1), (1,3,2), (1,0,1)\}$ is a basis for $\mathbb{Z}^3$.  Then
\begin{align*}
\frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{\langle(1,1,1),(1,3,2)\rangle} \cong \frac{\mathbb{Z} v_1 \oplus \mathbb{Z} v_2 \oplus \mathbb{Z} v_3}{\mathbb{Z}v_1 \oplus \mathbb{Z}v_2} \cong \mathbb{Z} v_3 \cong \mathbb{Z} \, .
\end{align*}
Alternatively, consider the homomorphism
\begin{align*}
\varphi : \mathbb{Z}^3 &\to \mathbb{Z}\\
(x,y,z) & \mapsto -x - y + 2z \, .
\end{align*}
One can show that $\varphi$ is onto and $\ker \varphi  = \langle (1,1,1), (1,3,2) \rangle$, so we can apply the first isomorphism theorem.
