HMMT 2021 T8

For each positive real number $\alpha$, define

$$\lfloor\alpha \mathbb{N}\rfloor:=\{\lfloor\alpha m\rfloor \mid m \in \mathbb{N}\}$$

Let $n$ be a positive integer. A set $S \subseteq\{1,2, \ldots, n\}$ has the property that: for each real $\beta>0$,

$$\text { if } S \subseteq\lfloor\beta \mathbb{N}\rfloor \text {, then }\{1,2, \ldots, n\} \subseteq\lfloor\beta \mathbb{N}\rfloor$$

Determine, with proof, the smallest possible size of $S$.