Question 26
Let $p$ and $n$ be positive integers, with $p \ge 2$, and let $a$ be a real number such that $1 \le a \lt a + n \le p$. Prove that the set
$$
\{ \lfloor \log_2 x \rfloor + \lfloor \log_3 x \rfloor + \dots + \lfloor \log_p x \rfloor \mid x \in \mathbb{R},\ a \le x \le a + n \}
$$
has exactly $n + 1$ elements.
$$
\{ \lfloor \log_2 x \rfloor + \lfloor \log_3 x \rfloor + \dots + \lfloor \log_p x \rfloor \mid x \in \mathbb{R},\ a \le x \le a + n \}
$$
has exactly $n + 1$ elements.