1. ## Sum of quintuples

Let $\displaystyle p \geq 7$ be a prime. Show that

$\displaystyle \Big(\sum_{1 \leq a <b<c<d<e\leq p-1}abcde\Big) \equiv 0 \mod p$

2. $\displaystyle (x-1)(x-2)(x-3) \cdots(x-(p-1)) = x^{p-1}-1$ over $\displaystyle \mathbb{F}_p$

Using Vieta's relations, the question asks for the coefficient of $\displaystyle x^{p-6}$ which is zero when $\displaystyle p \ge 7$

3. That's it!