In this and the subsequent problems, you will be working in a different number system

Chapter 39, Problem 39.24

(choose chapter or problem)

In this and the subsequent problems, you will be working in a different number system. The goal is to illustrate that unique factorization of prime numbers is a special feature of the integers. We consider all numbers of the form a C b p 3 where a and b are integers. For example, 5 2 p 3 is a number in this system, but 1 2 is not. This number system is denoted Z p 3. That is, Z p 3 is the set Z p 3 D n a C b p 3 W a; b 2 Z o : Please do the following: a. Prove that if w; z 2 Z p 3, then w C z 2 Z p 3. b. Prove that if w; z 2 Z p 3, then w z 2 Z p 3. c. Prove that if w; z 2 Z p 3, then wz 2 Z p 3. d. Find all numbers w such that both w and w1 are in Z p 3.