Math Artifacts

A set $S \subseteq \mathbb{N}$ satisfies the following conditions:

(a) If $x, y \in S$ (not necessarily distinct), then $x + y \in S.$

(b) If $x$ is an integer and $2x \in S,$ then $x \in S.$

Find the number of pairs of integers $(a, b)$ with $1 \le a, b \le 50$ such that if $a, b \in S$ then $S = \mathbb{N}.$ (OMO Spring 2016)