"Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$" I have a question that says this:

Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups.

I would like to see how it is correctly answered. This is not homework; I'd just like to see a proof.

For convenience, let $\mathcal G = \mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$
Since $\text{ord}\langle(1,1,1)\rangle = 40$, then $\text{ord} 
(\mathcal G) = \dfrac{5 \times 4 \times 8}{40} = 4$
Using this we compute
 a:  5  4  8
 s:  1  1  1
 g:  1  1  1
 m:  5  4  8
 p:  5  1  8
 t:  1  4  1
 r:  1  4  1
 α:  1  0  0
 ω:  1  1  1
 β:  1  0  0

And conclude that every $(x,y,z) \in \mathcal G$ can be expressed uniquely as
$$(x,y,z) = \kappa(1,1,1) + \lambda(0, 1, 0)$$
where $0 \le \kappa \lt 40$ and $0 \le \lambda \lt 4$
Consequently, $\mathcal G \cong \mathbb Z_4$
