Forced divisibility

Write each number as an odd number times a power of $2$ and group by the odd part.

1. Every integer from $1$ to $24$ is an odd number times a power of $2$, and there are $12$ odd numbers in that range.
2. Numbers with the same odd part, such as $3$ and $12$, always divide one another, so each odd part forms one group.
3. A set of $13$ numbers has $2$ numbers in one group, so it contains a dividing pair.
4. The set $\{13,14,\dots ,24\}$ has $12$ numbers and no number divides another, because twice any member exceeds $24$, so the answer is $13$.

Answer: $13$