Question 81
Let $p \geq 3$ be a prime. For $j=1,2, \ldots, p-1$, let $r_{j}$ be the remainder when the integer $\frac{j^{p-1}-1}{p}$ is divided by $p$. Prove that
$$
r_{1}+2 r_{2}+\ldots+(p-1) r_{p-1} \equiv \frac{p+1}{2}(\bmod p)
$$
$$
r_{1}+2 r_{2}+\ldots+(p-1) r_{p-1} \equiv \frac{p+1}{2}(\bmod p)
$$