Explicit and Recursive Formulas for Geometric Sequences

Let $S$ be a subset of the integers $1,2,\dots,100$ such that $3x\notin S$ whenever $x\in S$. What is the largest possible size of $S$?

(Hint: Consider geometric sequences of the form $x,3x,9x,27x,\dots$.)

Source: AHSME 1990