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set of 15 coprime numbers out of 2011 contains a prime number

Source: Singapore Junior Math Olympiad 2012 2nd Round p5 SMO

March 25, 2020
number theoryprime numberscoprimeprime

Problem Statement

Suppose S={a1,a2,...,a15}S = \{a_1, a_2,..., a_{15}\} is a set of 151 5 distinct positive integers chosen from 2,3,...,20122 , 3, ... , 2012 such that every two of them are coprime. Prove that SS contains a prime number. (Note: Two positive integers m,nm, n are coprime if their only common factor is 1)
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