Question 99
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 $\beta \gt 0$,
$$
\text{if } S \subseteq \lfloor \beta \mathbb{N} \rfloor, \text{ then } \{1,2, \ldots, n\} \subseteq \lfloor \beta \mathbb{N} \rfloor
$$
Determine, with proof, the smallest possible size of $S$.
$$
\lfloor \alpha \mathbb{N} \rfloor := \{ \lfloor \alpha m \rfloor \mid m \in \mathbb{N} \}
$$
Let $n$ be a positive integer. A set $S \subseteq \{1,2, \ldots, n\}$ has the property that: for each real $\beta \gt 0$,
$$
\text{if } S \subseteq \lfloor \beta \mathbb{N} \rfloor, \text{ then } \{1,2, \ldots, n\} \subseteq \lfloor \beta \mathbb{N} \rfloor
$$
Determine, with proof, the smallest possible size of $S$.