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Subsets and divisibility

Source: JBMO Shortlist 2006

November 10, 2008
number theorygreatest common divisoralgorithmmodular arithmeticnumber theory proposed

Problem Statement

Let A A be a subset of the set {1,2,…,2006} \{1, 2,\ldots,2006\}, consisting of 1004 1004 elements. Prove that there exist 3 3 distinct numbers a,b,c∈A a,b,c\in A such that gcd(a,b) gcd(a,b): a) divides c c b) doesn't divide c c
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