Why does an odd number times an odd number equal an odd number?

Let's do some algebra. Assume that "m" and "n" are any integers. An even number is divisible by 2, so 2m or 2n would be even. An odd number is one that is not divisible by 2, so 2m + 1, or 2n + 1, are odd numbers.

Multiply those two odd numbers together: (2m+1)(2n+1) = 4mn + 2m + 2n + 1. Since the first three parts are even, the added 1 at the end makes the result odd.