Problem 381

(prime-k) factorial

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}.

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