# The Odd Numbers

A number is considered **odd** if it **cannot** be equally divided by the number [latex]2[/latex]. It follows that since it is not divisible by [latex]2[/latex], any odd numbers are also **not** multiples of [latex]2[/latex].

What do you think would happen if you add [latex]+1[/latex] to any even numbers? Try to perform some calculations in your head and observe what you get.

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

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 [latex]n = 2k[/latex]. We simply add 1 to it. That’s why we arrive at [latex]n = 2k + 1[/latex]. In fact, the remainder of an odd number when divided by [latex]2[/latex] is always [latex]+1[/latex].

## General Form of an Odd Number

**DEFINITION:** The number [latex]\large{n}[/latex] is an odd number if it can be expressed as [latex]\large{2k+1}[/latex] where [latex]\large{k}[/latex] is just another integer.

## Examples of Odd Numbers Written in General Form

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

[latex]\color{red}\LARGE n = 2k + 1[/latex]

- [latex] – 109 \to – 109 = 2\left( { – 55} \right) + 1[/latex]

- [latex] – 13 \to – 13 = 2\left( { – 7} \right) + 1[/latex]

- [latex] – 5 \to – 5 = 2\left( { – 3} \right) + 1[/latex]

- [latex] – 1 \to – 1 = 2\left( { – 1} \right) + 1[/latex]

- [latex]7 \to 7 = 2\left( 3 \right) + 1[/latex]

- [latex]23 \to 23 = 2\left( {11} \right) + 1[/latex]

- [latex]49 \to 49 = 2\left( {24} \right) + 1[/latex]

- [latex]101 \to 101 = 2\left( {50} \right) + 1[/latex]

- [latex]377 \to 377 = 2\left( {188} \right) + 1[/latex]

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

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