The set $S = \{1, 2, 3, \ldots , 49, 50\}$ contains the first $50$ positive integers. After the multiples of 2 and the multiples of 3 are removed, how many integers remain in the set $S$?

To find a segment parallel to another segment and through a given point, fold a piece of paper so the fold goes through the point and the pieces of the segment on either side of the fold match up.