# 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 $$\dbinom{xp-1}{p} - \lfloor \dfrac{xp-1}{x} \rfloor \equiv r \pmod{p}$$?

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