Question 97
Let $p$ be an odd prime number and $x_{1}, \ldots, x_{p}$ integers.
Suppose that for every integer $k \geqslant 1$, we have
$$
p \mid x_{1}^{k}+\cdots+x_{p}^{k}
$$
Prove that the integers $x_{1}, \ldots, x_{p}$ are all congruent modulo $p$.
Suppose that for every integer $k \geqslant 1$, we have
$$
p \mid x_{1}^{k}+\cdots+x_{p}^{k}
$$
Prove that the integers $x_{1}, \ldots, x_{p}$ are all congruent modulo $p$.