= Solution
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 $r\geq0$ 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 $\{2^r d:r\geq0\}$. \b[The required representative set is $\boxed{A=\{1,3,5,\ldots\}}$.]
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