2025 USAJMO Problems/Problem 6

Problem

Let $S$ be a set of integers with the following properties:

$\bullet$ $\{ 1, 2, \dots, 2025 \} \subseteq S$ .

$\bullet$ If $a, b \in S$ and $\gcd(a, b) = 1$ , then $ab \in S$ .

$\bullet$ 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.

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2025 USAJMO (Problems • Resources)
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