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Set where no two elements divide each other

Source: China Mathematical Olympiad 2016 Problem 4

December 17, 2015
number theory

Problem Statement

Let n≥2n \geq 2 be a positive integer and define kk to be the number of primes ≤n\leq n. Let AA be a subset of S={2,...,n}S = \{2,...,n\} such that ∣A∣≤k|A| \leq k and no two elements in AA divide each other. Show that one can find a set BB such that ∣B∣=k|B| = k, A⊆B⊆SA \subseteq B \subseteq S and no two elements in BB divide each other.
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