Question 147
Let $p \geq 3$ be a prime number. Let $M$ denote the number of tuples $(x_1, x_2, x_3, x_4, x_5)$ of positive integers that satisfy the following conditions:
(1) $p \mid x_1^4 + x_2^4 + x_3^4 + x_4^4 + x_5^4$;
(2) $1 \leq x_1, \dots, x_5 \leq p$.
Find the remainder when $M$ is divided by $p$.
(Bilegdemberel Bat-Amgalan)
(1) $p \mid x_1^4 + x_2^4 + x_3^4 + x_4^4 + x_5^4$;
(2) $1 \leq x_1, \dots, x_5 \leq p$.
Find the remainder when $M$ is divided by $p$.
(Bilegdemberel Bat-Amgalan)