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Source: Iranian Third round 2015 - Number theory exam - Problem 2

September 6, 2015
Irannumber theory

Problem Statement

M0⊂NM_0 \subset \mathbb{N} is a non-empty set with a finite number of elements. Ali produces sets M1,M2,...,Mn M_1,M_2,...,M_n in the following order: In step nn, Ali chooses an element of Mn−1M_{n-1} like bnb_n and defines MnM_n as Mn={bnm+1∣m∈Mn−1}M_n = \left \{ b_nm+1 \vert m\in M_{n-1} \right \} Prove that at some step Ali reaches a set which no element of it divides another element of it.
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