We know $R=\{a+bi\sqrt{5}: a,b \in \mathbb{Z}\}$ is not a UFD because, for example, you can factor
$6=(1+i\sqrt{5})(1-i\sqrt{5})=(2)(3)$
and these are two distinct factorizations into irreducibles. This is can be shown using the norm $N:R \to \mathbb{N}$ defined $N(a+bi\sqrt{5})=a^2+5b^2$ and the fact that it's multiplicative, i.e. $N(rs)=N(r)N(s)$.
Since $R$ is not a UFD, $R$ is not a Euclidean domain either. How do we find an example to prove that our norm is not a Euclidean function? That is, how do we find a pair $a,b \in R$ for which there does not exist a pair $q,r \in R$ such that $b=aq+r$
with $N(r)
I would guess that the above factorization I gave is a place to start, but having looked at that for a bit, I still don't see where to begin.
This is a problem I'm working on in reviewing for a test, not a current homework problem.