I've asked a similar question: Computing Quotient Groups $\mathbb{Z}_4 \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$, $\mathbb{Z} \times \mathbb{Z}_{6}/ \langle (1, 2) \rangle$
But now I want to compute a quotient group involving a direct product in which every direct factor is infinite. For example $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} / \langle(1, 1, 1)\rangle$ or $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} / \langle(3, 3, 3)\rangle$. Is there a better approach than just looking for a homomorphism?
Thanks in advance.