Question 40
Given a prime number $p$, $p \ge 3$ and $d$ a square free number (that is $d$'s prime decomposition contains no repeated factors), find the number of the elements of the set
$$
A_p = \{x = \{n\sqrt{d} + \frac{n}{p}\} - \{n\sqrt{d}\} \mid n \in \mathbb{N}\},
$$
where $\{a\}$ denotes the fractional part of the real number $a$.
$$
A_p = \{x = \{n\sqrt{d} + \frac{n}{p}\} - \{n\sqrt{d}\} \mid n \in \mathbb{N}\},
$$
where $\{a\}$ denotes the fractional part of the real number $a$.