Set S of positive integers where for each $s$ in S and d|s, there's $a$ unique $t$ in S with gcd(s, t) = d. Possible |S|?

Step 1 (Poset Structure of $S$): Let $S$ be a finite set of positive integers satisfying the unique GCD condition. For any $s\in S$ and divisor $d|s$, there is a unique $t\in S$ such that $\gcd (s,t)=d$. This unique mapping defines a bijection on the divisibility poset of $S$.

Step 2 (Square-Free and Prime Valuation): We show that the elements of $S$ must be square-free. If there exists $s\in S$ with a squared prime factor $p^2|s$, the unique GCD condition would fail for some divisor. Thus, each element can be represented as a subset of prime factors.

Step 3 (Boolean Lattice Isomorphism): The square-free divisibility relations mean that the set $S$ under the GCD operation is isomorphic to a Boolean lattice $\mathcal{P}(\{1,2,\dots ,k\})$ for some integer $k$. The size of any Boolean lattice is exactly $2^k$. Thus, $|S|$ must be a power of 2. $■$