Fermat and Wilson's Theorem

1. If and are distinct primes prove that for any integer a, Use Fermat’s theorem 2. Show that if and are both primes, then 4[ (mod Use Wilson’s theorem. 3. Let be an odd prime. Prove that if g is primitive root modulo and (mod is not Use the binomial expansion See attached file for full problem description.

Find the simultaneous solutions of the following congruences

1. Prove that gcd (a, lcm[b, c]) = lcm[gcd(a,b), gcd(a,c)]. 2. Find the simultaneous solutions of the following congruences: 2x ≡ 1(mod 5) 3x ≡ 9 (mod 6) 4x ≡ 1 (mod 7) 5x ≡ 9 (mod 11)

Properties of groups

If G is a group, then (1) the identity element of G is unique, (2) every a belongs to G has a unique inverse in G.

Show that an r-cycle is an even permutation if and only if r is odd.

1. If alpha is an r-cycle, show that alpha^r = (1). [There's a hint that If alpha = (i sub 0 ... i sub r-1), show that alpha ^k(i sub 0) = i sub k.] 2. Show that an r-cycle is an even permutation if and only if r is odd. 3. If alpha is an r-cycle and 1

Noncyclic Group Order 4.

Noncyclic Group Order 4. See attached file for full problem description.