"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.

I'm assuming that $\langle(1,1,1)\rangle$ means the subgroup generated by $(1,1,1)$, or in other words $\{(k\bmod 5,k\bmod 4,k\bmod 8)\mid k\in\mathbb Z\}$.
In that case we can see that each of the cosets that make up the quotient must contain an element of the form $(0,x,0)$. Namely, assume that $(a,b,c)$ is some element of the cosets; then by the Chinese Remainder Theorem we can find $k$ such that $a\equiv k\bmod 5$ and $c\equiv k\bmod 8$. Subtracting $(a,b,c)-(k,k,k)$ gives us an alement o the form $(0,x,0)$.
On the other hand, $(0,x,0)$ is can only be zero in the quotient when $x\equiv 0\bmod 4$ (because if $(0,x,0)\equiv(k,k,k)$ then $k\equiv 0\bmod 8$ and therefore $k\equiv 0\bmod 4$.
So the quotient group is $\mathbb Z_4$.
What this has to do with the structure theorem I don't know, though. Perhaps you're supposed to start by rewriting it to $\mathbb Z_4\times\mathbb Z_{40}/\langle(1,1)\rangle$?
