Prime Numbers Congruence

Theorem: If p is prime, and a + b = p - 1 for a, b positive integers, then a!*b! "is congruent to" (-1)^(b + 1) (mod p)

a.) Illustrate two examples (w/ primes that are greater than 5) showing the theorem above.

b.) Prove the above theorem.

WORK:

a.) Showing it:

I'm assuming you pick a prime. So, let's pick 7 and 11.

a + b = 7 - 1 = 6

a!*b! "is congruent to" (-1)^(b + 1) (mod 7) ... not sure where to go from here.

And similarly for 11:

a + b = 11 - 1 = 10

a!*b! "is congruent to" (-1)^(b + 1) (mod 11)

And no clue how to do part b.

Thanks!

It reminds me of Gauss' Lemma. Here is the proof below.