# 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. Here it is!

**Let’s agree!**

Just for this lesson, we need to have an agreement with regard to the remainders to ONLY allow non-negative integers. If that is our restriction, the allowable remainders will be zero (0) and any positive integers.

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

Well, it’s true that an even number added by 1, or an even 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 that 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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