Given $x$ and $y$ in $\mathbb{Z}[i]$, find $q$ and $r$ such that $x=qy+r$.

Question

Find $q, r \in \mathbb{Z}[i]$ such that:

1. $1 + 5i = (1 + 2i)q + r$ with $|r| < 2$,

2. $1 + 5i = (2i)q + r$ with $|r| < 2$.

My only train of thought is that $r = 1+0i$, $0+i$ or $0$.

Otherwise I'm quite lost.

Edit: Is the 2nd part possible at all? I cant figure out anything that works.

Answer 1

Hint for (1). Show that the problem is equivalent to finding a lattice point $q$ that is within a distance of $2/\sqrt{5}$ from $(1 + 5i)/(1 + 2i)$.