USAJMO 2025 Question 6

Let $S$ be a set of integers such that (i) $\{1,2,\ldots,2025\}\subseteq S$; (ii) if $a,b\in S$ and $\gcd(a,b)=1$, then $ab\in S$; (iii) if $s\in S$ and $s+1$ is composite, then all positive divisors of $s+1$ belong to $S$. Prove that $S$ contains all positive integers.

Proposed by Enrique Treviño.

Manual essay; 1 editorial point, not official contest scoring.