(a) For a prime p of the form 4k + 3, prove that eitherhence, [(p− l)/2]! satisfies the qu...

(a) For a prime p of the form 4k + 3, prove that either

hence, [(p− l)/2]! satisfies the quadratic congruence x2 ≡ 1 (mod p).

(b) Use part (a) to show that if p = 4k + 3 is prime, then the product of all the even integers less than p is congruent modulo p to either 1 or − 1.

[Hint: Fermat’s theorem implies that 2(p−1)/2 ≡ ±1 (mod p).]