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Iberoamerican Olympiad 2013 - Problem 1

Source: http://oim2013.opm.org.pa/pdfs/examen_pt.pdf

August 13, 2014
number theoryleast common multiplenumber theory proposed

Problem Statement

A set SS of positive integers is said to be channeler if for any three distinct numbers a,b,c∈Sa,b,c \in S, we have a∣bca\mid bc, b∣cab\mid ca, c∣abc\mid ab.
a) Prove that for any finite set of positive integers {c1,c2,…,cn} \{ c_1, c_2, \ldots, c_n \} there exist infinitely many positive integers kk, such that the set {kc1,kc2,…,kcn} \{ kc_1, kc_2, \ldots, kc_n \} is a channeler set.
b) Prove that for any integer n≥3n \ge 3 there is a channeler set who has exactly nn elements, and such that no integer greater than 11 divides all of its elements.
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