Question 51
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 $1 \lt k \lt m$, the number $km + 1$ is also an element of $M$.
Prove that $M = \mathbb{Z}_{\ge 1}$.
(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 $1 \lt k \lt m$, the number $km + 1$ is also an element of $M$.
Prove that $M = \mathbb{Z}_{\ge 1}$.