1. Let p be an odd prime. Let a be an integer such that x2 = a (mod p) has that a (mod p has exactly two solutions for all a solution. Then prove positive integers k

Fermat's Little Theorem says that for all natural numbers p and integers x, if p is prime then x² = x mod p. (i) Show that the statement is wrong, if we drop the assumption that p is prime. [Remember: All you need to do is to give a counterexample.] (ii) Show, by giving an example, that if p is not prime, then x² = x modp can still be true for some integers x with x ‡ −1, 0, 1.

3. Find all integer solutions to the pair of congruences (if any) x = 1 (mod 2) * = 2 (mod3) x = 3 (mod 5) x = 4 (mod 11)

Let p be an odd prime and let a and b be integers with ab #0(mod p). Show that ax² = b[mod p) has a solution if and only if ()) = 1.

Find a number a between 0 and 86 such that 71122 = a (mod 87).

Find the smallest positive integer k such that 2k = 1 (mod 17).

Prove that if p > 2 is a prime, then (1 2) and (1 2... p-1 p) generate Sp.

Both part (a.) and (b.) please.

Fermat's Little Theorem says that for all natural numbers p and integers x, if p is prime then xº = x mod p. (i) Show that the statement is wrong, if we drop the assumption that p is prime. [Remember: All you need to do is to give a counterexample.] (ii) Show, by giving an example, that if p is not prime, then xP = x mod p can still be true for some integers x with x -1, 0, 1.