Similar to Wilson's Theorem

Suppose that m is an int. and has primitive roots; then, we know that the product of the pos. integers less than m and also relatively prime to m is congruent to -1 (mod m). (Notice that if m were prime, this would just be Wilson’s Theorem).

a.) Give an example of this theorem for a m (a compositive number) that is greater than 10

b.) Give an example that shows the result of this theorem will not always be true if m doesn't have primitive roots.

c.) Prove the theorem.

HINT: for all primes p, if gcd(a,p) = 1, then a^((p-1)/2) = +/- 1 (mod p)

This idea then leads to this next theorem which you may want to use in your proof: Theorem: If m (composite or prime) has a primitive root r,

then r^(phi(m)/2) = -1 (mod m)