# How many residues?

**Number Theory**Level 4

Let \( x \) be a positive integer, \( p \) be an odd prime and \( r \) be a positive integer where \( 0 \le r \le p-1 \). As \( x \) ranges over all positive multiples of \( p \), how many values of \( r \) are there which satisfy \( \dbinom{xp-1}{p} - \lfloor \dfrac{xp-1}{x} \rfloor \equiv r \pmod{p} \)?