**Odd numbers** are the numbers that cannot be divided into two separate groups evenly. These numbers are not completely divided by 2, which means there is some remainder left after division.

## What are Odd Numbers?

Odd numbers are defined as any number which cannot be divided by two is termed as an odd number. In other words, a number of form 2k+1, where k ∈ Z (i.e. integers) are called **odd numbers**. It should be noted that numbers or set of integers on a number line can either be odd or even. A few more key points:

- An odd number is an integer which is not a multiple of 2.
- If these numbers are divided by 2, the result or remainder should be a fraction or 1.
- In the number line, 1 is the first positive odd number.

## Odd Numbers Chart

The odd number chart consist of odd numbers from 1 to 100. You can also practice writing the odd numbers from 1 to 1000 in your notebook.

## Odd Numbers List

There are 25 odd numbers from 1 to 50 while there are 50 in between 1 to 100. In case of numbers from 1 to 1000, there are 500 odd numbers and 500 even numbers. A few odd numbers list include numbers like:

- -5, -3, -1, 1, 3, 5 , 7 , 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, etc.

Number Range | No. of Odd Numbers |
---|---|

1 to 50 | 25 |

1 to 100 | 50 |

1 to 1000 | 500 |

## Properties of Odd Numbers

There are four main properties of odd numbers which are related to their addition, subtraction, multiplication, and division.

### Adding Two Odd Numbers

Any odd number added to another odd number always gives an even number. This statement is also proved below.

Odd + Odd = Even |

**Proof:**

Let two odd numbers be a and b.

These numbers can be written in the form where

a = 2k_{1} + 1

and b = 2k_{2} + 1 where k_{1}, k_{2} ∈ Z

Adding a + b we have,

(2k_{1} + 1) + (2k_{2} + 1) = 2k_{1} + 2k_{2} + 2 = 2(k_{1} + k_{2} + 1) which is surely divisible by 2.

### Subtracting Two Odd Numbers

When an odd number is subtracted from an odd number, the resultant number will always be an even number. This is similar to adding two odd numbers where it was proved that the resultant was always an even number.

Odd – Odd = Even |

### Multiplication of Two Odd Numbers

If an odd number is multiplied by another odd number, the resulting number will always be an odd number. A proof of this is also given below.

Odd × Odd = Odd |

Let two odd number be a and b. These numbers can be written in the form where

a = 2k_{1} + 1 and b = 2k_{2} + 1 where k_{1} , k_{2} ∈ Z

Now, a × b = (2k_{1} + 1)(2k_{2} + 1)

So, a × b = 4k_{1 }k_{2} + 2k_{1} + 2k_{2} + 1

The above equation can be re-written as:

a × b = 2(2k_{1} k_{2} + k_{1} + k_{2}) + 1 = 2(x) + 1

Thus, the multiplication of two odd number results is an odd number.

### Division of Two Odd Numbers

Division of two odd numbers always result in Odd number if and only if the denominator is a factor of the numerator, or else the number result in decimal point number.

Odd ⁄ Odd = Odd |

**In short:**

Operation | Result |
---|---|

ODD + ODD | EVEN |

ODD – ODD | EVEN |

ODD x ODD | ODD |

ODD / ODD
*denominator is a factor of the numerator |
ODD |

## Types of Odd Numbers

There are 2 main types of odd numbers which are **consecutive odd numbers** and **composite odd numbers**.

### Consecutive Odd Numbers

If ‘a’ is an odd number, then ‘a’ and ‘a + 2’ are called consecutive odd numbers. A few examples of consecutive odd numbers can be

- 15 and 17
- 29 and 31
- 3 and 5
- 19 and 21 etc.

Even for negative odd numbers, consecutive ones will be

- -5 and -3
- -13 and -11, etc.

### Composite Odd Number

A composite odd number is a positive odd integer which is formed by multiplying two smaller positive integers or multiplying the number with one. The composite odd numbers up to 100 are: 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99.

## Example Questions

**Example 1:** Find the sum of the smallest and the largest 3 digit odd number and also prove that it is divisible by 2.

**Solution**:

Smallest 3 digit odd number = 101

Largest 3 digit odd number = 999

Sum of both the numbers = 101 + 999 = 1100

The number 1100 is divisible by 2 (as per divisibility rule of 2).

This proves that the number is even.

**Example 2**: Are the following numbers odd?

- 25
- 15 + 13
- 32 – 37

**Solution**:

- 25 is not divisible by 2, so odd number.
- 15 + 13 = 28, divisible by 2, not an odd number
- 32 – 37 = -5, is an odd number

### Odd Number Worksheet for Practice

- Is 7 even or odd?
- How do you determine if a number is odd or even?
- Mention all the odd numbers which are greater than 60 and smaller than 120.
- List all the odd numbers which are greater than -4 and smaller than 20.
- Is zero an odd number? Why?

## Frequently Asked Questions on Odd Numbers

### What are all the Odd Numbers?

The numbers which are not divisible by 2, evenly, are called Odd Numbers. For Example: 3, 5, 17, 19, 21, etc.

### How do you determine if a number is odd or even?

If a number is evenly divided by 2 then it’s an even number, otherwise, it’s an odd number. We can also say when we divide a number by 2 and there is some remainder left, which is not divisible again by 2, then its an odd number.

For a greater number which is in ten thousands or millions, check the number at the unit place. If unit place carries an even number, that means the whole number is even else its an odd one.

### Is zero an odd number? Why?

No, zero is not an odd number but an even number, because, when we divide 0 by 2, it gives us quotient as 0 and also there is no remainder left after division. So, 0 is evenly divided by 2.

0 ÷ 2 = 0

### Is 37 an Odd number or Even?

As we can see, the unit place of 37 consists of an odd number i.e. 7, which is not evenly divisible by 2, therefore, its an odd number.