1.4-6: Use a direct proof to show that the product of 2 odd integers is odd.

Section 1.4 Problem 6.

Proof. Assume that x and y are odd integers. By definition of odd, there are integers a and b such that x = 2a + 1 and y = 2b + 1. By substitution

xy = (2a + 1) (2b + 1)

= 4ab + 2a + 2b + 1

= 2(2ab + a + b) + 1.

Since 2, a, and b are integers and integers have closure with respect to multiplication and addition, then 2ab + a + b is an integer. By definition of odd, it follows that the product of xand y is odd. ☐