Question 180
Let $S$ be a set of positive integers with the following properties:
* $\{1, 2, \dots, 2025\} \subseteq S$.
* If $a, b \in S$ and $\gcd(a, b) = 1$, then $ab \in S$.
* If for some $s \in S$, $s+1$ is composite, then all positive divisors of $s+1$ are in $S$.
Prove that $S$ contains all positive integers.
* $\{1, 2, \dots, 2025\} \subseteq S$.
* If $a, b \in S$ and $\gcd(a, b) = 1$, then $ab \in S$.
* If for some $s \in S$, $s+1$ is composite, then all positive divisors of $s+1$ are in $S$.
Prove that $S$ contains all positive integers.