The product of two even numbers is even.

Let m and n be any integers so that 2m and 2k are two even numbers.

The product is 2m(2k) = 2(2mk), which is even.

Things to think about:

Why didn’t I just show you by using any two even numbers like the number 4 and the number 26?

Why did I change from "m" to "k" ? Are they really different numbers or could they be the same?

Why did I specifically say that m and k were integers?

The product of two odd numbers is an odd number.

Let m and k be any integers. This means that 2m+1 and 2k+1 are odd numbers.

The product is 4mk + 2m + 2k + 1 (hint: I used FOIL) which can be written as

2 ( 2mk + m + k ) + 1 which is an odd number.

Things to ask yourself:

Why isn’t a demonstration of this fact using 3 and 7 enough to be considered a proof?

Is it ok to re-use m and k again in this proof? Why or why not?

What definitions lie behind these proofs?