Problem 1. (10 points) Let p be a prime number. In class we proved that every non-zero element of Zp has a multiplicative inverse. Since 1.11 (mod p) it is obvious that 1-1 1 (mod p). In other words, 1 is its own inverse, and we say that 1 is a self-inverse mod p. For each non-zero number in Z5 compute its inverse mod 5. Which numbers are self-inverses mod 5 For each non-zero number in Z11 compute its inverse mod 11. Which numbers are self-inverses mod 11? a. b. c. Prove that the only self-inverses mod p in Zp are 1 andp 1. To get started, note that if k is a self-inverse then k2 1 (mod p) Starting with this congruence, use the fact that k2-1- (k-1). (k 1) to complete your proof. (Extra Credit) For any natural number n, the factorial function n! is defined as d. n! n(n - 1(n2)1 Prove that for every prime number p, (p-1)! -1 (mod p) Hint: Consider every number in the product and use the result of part (c)