Let a and b be any two odd integers. We wish to show that ab is also an odd integer.) 2. There are two integers m and n such that a = 2m + 1 and b = 2n + 1. 3. ab = (2m + 1)(2n + 1) 4. (2m + 1)(2n + 1) = 4mn + 2m +2n +1 5. 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 6. Since m and n are integers, 2mn + mun is also an integer 7. Thus, ab is an odd integer. Distribute. Apply the definition of odd integers for the first time. Apply the definition of odd integers for the second time. Integers are closed under multiplication and addition To prove something about all two odd integers, we assume that we have two arbitrary odd integers. Factor 2 from terms where it occurs. Substitute 2m + 1 for a and 2n +1 for b in the product ab.