The Definition of an Odd Number

An odd number is an integer not divisible by 2 without having a remainder. Odd numbers end in 1, 3, 5, 7, and 9. In other words, an integer is considered an odd number if dividing it by 2 does not result in an integer.

An integer is a whole number. Examples of integers are 0, 5013, and -44. Examples of non-integers are 3/5, 1.3, and -√11. Non-integers are not considered odd or even. They do not follow the requirements of either category.

How to Prove Odd Numbers are Odd

Here's a look at the single digit odd numbers being divided by two. Remember, these are integers that result in a non-integer when dividing by two:

• ¹⁄₂ = 0.5
• ³⁄₂ = 1.5
• ⁵⁄₂ = 2.5
• ⁷⁄₂ = 3.5
• ⁹⁄₂ = 4.5

We can keep going up and up the list of odd numbers and the result of dividing by two will always be 0.5 ± some integer.

Odd Numbers Example Problems

Let's go through a couple of example problems together to reinforce our ability to identify odd numbers.

Example Problem 1

Is 24 an odd number?

Solution:

24 is an integer. To see if it is an odd number, let’s check with both methods.

The last digit is 4, so it must not be an odd number. Dividing it by 2, we get 24/2 = 12, which has no remainder/is an integer result.
No, 24 is not an odd number.

Example Problem 2

Is 37 an odd number?

Solution:

37 is an integer. To see if it is an odd number, let’s check with both methods.

The last digit is a 7 so we know it must be an odd number. Dividing it by 2, we get 37/2 = 18.5, which has a remainder/is a non-integer result.
Yes, 37 is an odd number.