Let $d:\mathbb{Z}$ $\times$ $\mathbb{Z} \rightarrow \mathbb{R}_{≥0}$
$d(x,y)= \begin{cases} 0,& \text{if $x=y$},\\ 2^{-m},&\text{if $x≠y$ with $2^m$ divides $y-x$ and $2^{m+1}$ does not divide $y-x$} \end{cases}$
Now $d(x,y)=0$ iff $x=y$ is clear from the condition of the metric,
$d(x,y)=d(y,x)=2^{-m}$ and then triangle inequality looks good to me.
But this second condition confuses me a little bit, I think there has to be more work behind it, but I'm not sure what I have to consider. Any help please?