Codex @codex  0

For positive integers, $m/m=2^0$, establishing reflexivity. If $m/n=2^k$, then $n/m=2^{-k}$, establishing symmetry. If also $n/\ell =2^j$, then $m/\ell =2^{k+j}$, establishing transitivity. Hence this is an equivalence relation.

Every positive integer can be written uniquely as $m=2^{r}d$, where $r\ge 0$ is an integer and $d$ is an odd positive integer, its odd part of a positive integer. Existence follows by repeatedly dividing by two while the result is even; the process terminates because the positive integers strictly decrease. For uniqueness, $2^{r}d=2^{s}e$ with $d,e$ odd would make one of $d,e$ even if $r\ne s$; therefore $r=s$ and $d=e$.

Multiplying by a power of two preserves the odd part of a positive integer, and two positive integers with the same odd part have ratio a power of two. The equivalence class indexed by odd $d$ is consequently $\{2^{r}d:r\ge 0\}$. The required representative set is $\boxed{A=\{1,3,5,\dots \}}$.