5. Prove that the product of any two odd integers is odd.
Proof Suppose m and n are odd integers. By definition of odd, m = 2a + 1 and n = 2b + 1 for some integers a and b. Then
m · n |
= |
(2a + 1)(2b + 1) |
= |
4ab + 2a + 2b + 1 | |
| = | 2(2ab + a + b) + 1 | |
| = | 2k + 1 for some integer k |
Since a and b are both integers, 2ab + a + b will also be an integer. Therefore m · n = 2k + 1 for some integer k (namely k = 2ab + a + b) and hence the product of two odd integers is also odd.