Consider the ring of integral quaternions, $\mathbb{I}$, with norm-like function $N(a+bi+cj+dk) = a^2+b^2+c^2+d^2$.
Let $z,w \in \mathbb{I}$, with $w \neq 0$. Prove that $\exists q, r \in \mathbb{I}$ such that $z = qw + r$, with $N(r) < N(w)$.
We already know such things like: $N$ is multiplicative, there are precisely 24 units in $\mathbb{I}$ every element in $\mathbb{I}$ is associate to an element with all coefficients integers.