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GCD's implying GCD's

Source: European Mathematical Cup 2012, Junior Division, Problem 2

July 27, 2013
number theorygreatest common divisorfunctionprime numbersnumber theory proposed

Problem Statement

Let SS be the set of positive integers. For any aa and bb in the set we have GCD(a,b)>1GCD(a, b)>1. For any aa, bb and cc in the set we have GCD(a,b,c)=1GCD(a, b, c)=1. Is it possible that SS has 20122012 elements? Proposed by Ognjen Stipetić.
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