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Maximal size set of S with 3 conditions

Source: Chinese Mathematical Olympiad 2003 Problem 2

February 18, 2012
number theoryprime numberscombinatorics proposedcombinatorics

Problem Statement

Determine the maximal size of the set SS such that: i) all elements of SS are natural numbers not exceeding 100100; ii) for any two elements a,ba,b in SS, there exists cc in SS such that (a,c)=(b,c)=1(a,c)=(b,c)=1; iii) for any two elements a,ba,b in SS, there exists dd in SS such that (a,d)>1,(b,d)>1(a,d)>1,(b,d)>1.
Yao Jiangang
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