When a is a factor of a number b, then b is divisible by a. For example, 3 is a factor of 12, so 12 is divisible by 3. We will use this concept in the proof in this post.
Notice that the sum of the following consecutive odd numbers.
11 + 13 = 24
9 + 11 = 20
25 + 27 = 52
101 + 103 = 204
Notice that the sum is divisible by 4. Now, we can make a conjecture that the sum of two consecutive numbers is divisible by 4.
Theorem: The sum of two consecutive odd numbers is divisible by 4.
Explanation and Proof
An even number is divisible by 2, so it can be represented by 2n, where n is an integer. If we add 1 to an even number, then it will be odd. Therefore, an odd number can be represented as 2n + 1.
If 2n + 1 is an odd number, then the next odd number is 2n + 3 (Why?). Therefore, the sum of two consecutive odd numbers can be represented by
(2n + 1) + (2n + 3) = 4n + 4.
Factoring, we have 4(n + 1). This means that 4 is a factor of 4n + 4. This means that 4n + 4 is divisible by 4. This implies that the sum of two consecutive odd numbers is divisible by 4.