AT-OEMO-FED1 2018 Problem 4

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)