We can write each positive integer $ n$ in a unique way as $ n = 2^k\cdot q$ with $ k,q\in\mathbb{N}$ and $ q$ odd. If two numbers have the same $ q$, one is divisibly by another. Since there are only $ 50$ odd numbers in $ [1,100]$, the requested number is at most $ 50$. But the set $ \{51,52,53,...,100\}$ is an example of such a set with $ 50$ elements, so the maximum number of elements is $ 50$. I think, by the way, that the source of this problem is Erdös.