Evaluate the proof of the following result.Result Let a, b Z. Then a b is even if and

Chapter 3, Problem 3.40

(choose chapter or problem)

Evaluate the proof of the following result.Result Let a, b Z. Then a b is even if and only if a and b are of the same parity.Proof We consider two cases.Case 1. a and b are of the same parity. We now consider two subcases.Subcase 1.1. a and b are both even. Then a = 2x and b = 2y, where x, y Z. Thena b = 2x 2y = 2(x y). Since x y is an integer, a b is even.Subcase 1.2. a and b are both odd. Then a = 2x + 1 and b = 2y + 1, where x, y Z. Thena b = (2x + 1) (2y + 1) = 2(x y). Since x y is an integer, a b is even.Case 2. a and b are of opposite parity. We again have two subcases.Subcase 2.1. a is odd and b is even. Then a = 2x + 1 and b = 2y, where x, y Z. Thena b = (2x + 1) 2y = 2(x y) + 1. Since x y is an integer, a b is odd.Subcase 2.2. a is even and b is odd. Then a = 2x and b = 2y + 1, where x, y Z. Thena b = 2x (2y + 1) = 2x 2y 1 = 2(x y 1) + 1. Since x y 1 is an integer, a b is odd.