Here's my attempt as I think I vaguely remember something similar:

We have $\phi : \mathbb{Z} \to \mathbb{Z}[i]/(7+5i)$ by $\phi(n) = n + (7+5i)$. I would like now to prove that $\ker\phi = \langle 74 \rangle$ and so by the first isomorphism theorem $$\mathbb{Z}[i]/(7+5i) \cong \mathbb{Z} / \langle 74 \rangle= \mathbb{Z}_{74}.$$

Questions:

- How do we prove that $\ker \phi = \langle 74 \rangle $
- Is the last equals sign correct?