# Order Of A Polynomial Shuffle

**Number Theory**Level 5

For how many possible values of \(n\), can one find a polynomial with integer coefficients \(f(x)\) and pair-wise distinct integers \(x_1,x_2,...,x_n,\) such that \(0\leq x_i \leq 24\) for all \(i\) and \[\begin{cases}f(x_1)\equiv x_2 \pmod {25}\\ f(x_2)\equiv x_3 \pmod {25}\\ \ ...\\ f(x_n)\equiv x_1 \pmod {25} \ ?\end{cases}\]