Question 100
Let $p$ be a prime number, and $f\left(x_{1}, \ldots, x_{n}\right)$ be a polynomial with integer coefficients of total degree less than $n$. Prove that the number of ordered $n$-tuples $\left(x_{1}, \ldots, x_{n}\right)$ with $0 \leq x_{i}\lt p$ such that $f\left(x_{1}, \ldots, x_{n}\right)$ is an integer multiple of $p$ is an integer multiple of $p$.