Prove ring of dyadic rationals is a Euclidean domain

Question

This is a question from page 105, Vinberg - A course in Algebra:

Prove that the ring $A$ of rational numbers of the form $ 2^{-n}m,~m \in \mathbb{Z},~n \in \mathbb{Z}_+ $, is a Euclidean domain.

Here $\mathbb{Z}_+$ contains 0. I try to find a norm $N \colon A \setminus \{0\} \to \mathbb{Z}_+$ that satisfies Euclidean algorithm. I suppose that $N$ has to satisfy the following conditions:

1. Since each rational number is an equivalent class, the norm $N$ must not depend on the representative of the equivalent class.

2. $N$ somehow measures the "distance" between elements in A and 0 (as we did for the case Gaussian integers $\mathbb{Z}[i]$) because we need to compare the norms of divisor and remainder.

However, I can't find any reasonable formula for $N$. Any ideas?

Answer 1

Each nonzero element in the ring is $2^ka$ where $k\in\Bbb Z$ and $a$ an odd integer. Define $N(2^ka)=|a|$,