24. (Division with remainder in $\boldsymbol{Z}[\sqrt{-1}]$) Let $\alpha_{j}=a_{j}+\mathrm{i} b_{j} \in \boldsymbol{Z}[\sqrt{-1}], a_{j}, b_{j} \in \boldsymbol{Z}$ $(j=0,1)$.
(i) Prove: There must exist $\eta_{1}, \alpha_{2} \in \boldsymbol{Z}[\sqrt{-1}]$, satisfying
$$\alpha_{0}=\eta_{1} \alpha_{1}+\alpha_{2}, \quad 0 \leqslant N\left(\alpha_{2}\right)<N\left(\alpha_{1}\right)$$

where $N[\alpha]=r^{2}+s^{2}, \alpha=r+s \sqrt{-1}, r, s \in Q$, here $N(\alpha)$ is called the norm of $\alpha$;
(ii) Are $\eta_{1}, \alpha_{2}$ in (i) unique? What is the maximum number of solutions?