# The Odd Numbers

An **odd number** is an integer that **cannot** be divided evenly or exactly by the number \color{red}2. Another way of looking at it, since an odd number is not divisible by 2, it implies that any odd numbers are not a multiple of 2.

For the sake of uniformity of our final answers, we will adopt a certain condition regarding the nature of our remainders.

**Let’s agree!**

Just for this lesson, we need to have an agreement to ONLY allow nonnegative integers when it comes to the remainders. If that is our restriction, the allowable remainders will be zero (0) and any positive integers.

What do you think would happen if you add +1 to any even numbers? Try to perform some calculations in your head and observe what you get. Do you obtain the same pattern when you take an even number then subtract it by 1?

Well, it’s true that an even number added by 1, or an even number subtracted by 1 will always yield an odd number.

Below are some examples to illustrate the case when an even number is increased by 1.

We can now describe the general form of an odd number. Notice, it’s very similar to the general form of an even number which is n = 2k. We simply add 1 to it. That’s why we arrive at n = 2k + 1. In fact, the remainder of an odd number when divided by 2 is always +1.

## General Form of an Odd Number

**DEFINITION:** The number \large{n} is an odd number if it can be expressed or written as \large{2k+1} where \large{k} is just another integer. Thus, \large{n} is odd if \large{n=2k+1} such that \large{k} is an integer.

## Examples of Odd Numbers Written in General Form

Let’s put it to the test if indeed an odd number \textbf{n} can be expressed in the form \textbf{n = 2k + 1} where \textbf{k} is also an integer.

\color{red}\LARGE n = 2k + 1- - 109 \to - 109 = 2\left( { - 55} \right) + 1

- - 13 \to - 13 = 2\left( { - 7} \right) + 1

- - 5 \to - 5 = 2\left( { - 3} \right) + 1

- - 1 \to - 1 = 2\left( { - 1} \right) + 1

- 7 \to 7 = 2\left( 3 \right) + 1

- 23 \to 23 = 2\left( {11} \right) + 1

- 49 \to 49 = 2\left( {24} \right) + 1

- 101 \to 101 = 2\left( {50} \right) + 1

- 377 \to 377 = 2\left( {188} \right) + 1

- 1,013 \to 1,013 = 2\left( {506} \right) + 1

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