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subset of pos. integers with 3 properties , prove that it is the set {1,2,3,...}

Source: 49th Austrian Mathematical Olympiad National Competition (Final Round) 28th April 2018 p4

May 25, 2019

number theorypositive integers

Problem Statement

Let

M

be a set containing positive integers with the following three properties: (1)

2018 \in M

. (2) If

m \in M

, then all positive divisors of m are also elements of

M

. (3) For all elements

k, m \in M

with

1 < k < m

, the number

km + 1

is also an element of

M

. Prove that

M = Z_{\ge 1}

.
(Proposed by Walther Janous)