## (prime-k) factorial

### Problem 381

Published on Saturday, 21st April 2012, 05:00 pm; Solved by 2028; Difficulty rating: 15%For a prime p let S(p) = (∑(p-k)!) mod(p) for 1 ≤ k ≤ 5.

For example, if p=7,

(7-1)! + (7-2)! + (7-3)! + (7-4)! + (7-5)! = 6! + 5! + 4! + 3! + 2! = 720+120+24+6+2 = 872.

As 872 mod(7) = 4, S(7) = 4.

It can be verified that ∑S(p) = 480 for 5 ≤ p < 100.

Find ∑S(p) for 5 ≤ p < 10^{8}.