Odd part of a positive integer ($n=2^{v_2(n)}d$)

The odd part of a positive integer $n$ is the unique odd positive integer $d$ obtained by repeatedly dividing $n$ by two. The process terminates because the positive integers decrease. If $2^{r}d=2^{s}e$ with $d,e$ odd, then $r>s$ would make $e$ even and $s>r$ would make $d$ even; hence $r=s$ and $d=e$. Two positive integers have the same odd part exactly when their ratio is an integral power of two. The odd positive integers therefore give representatives for the resulting equivalence classes.