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find sets with three conditions, closed under gcd and *

Source: IMOC 2021 N5

August 12, 2021
number theorygreatest common divisorSets

Problem Statement

Find all sets SS of positive integers that satisfy all of the following.
1.1. If a,ba,b are two not necessarily distinct elements in SS, then gcd⁡(a,b)\gcd(a,b), abab are also in SS. 2.2. If m,nm,n are two positive integers with n∤mn\nmid m, then there exists an element ss in SS such that m2∣sm^2\mid s and n2∤sn^2\nmid s. 3.3. For any odd prime pp, the set formed by moduloing all elements in SS by pp has size exactly p+12\frac{p+1}2.
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