Show that the product of is an even number.

This proof requires elementary knowledge in algebra and some manipulation of symbols. It is assumed that you already know or have proved that the sum of two even numbers is even, the sum of two odd integers is odd, and the sum of an odd number and an even number is odd. This proof is designed to be read by middle school and high school students, so it is written with details.

**Proof 1**

The numbers and are consecutive numbers. There are two cases possible, the smaller number is odd or even.

If the smaller number is odd, then is even. Now, odd multiplied by even is even.

For the second case, if the smaller number is even, then the second number is odd. Now, even multiplied by odd is always even.

Therefore, in any case the product of is even. Continue reading…