# Odd Numbers

Odd numbers can be defined as the numbers that cannot be divided into two separate groups evenly. Here, all the concepts related to odd numbers like odd numbers definition, examples, etc. are covered. The key odd number concepts included here are as follows:

• What are Odd Numbers?
• Odd Numbers Chart
• Odd Numbers List 1 to 100
• Consecutive Odd Numbers
• Composite Odd Number
• Properties of Odd numbers
• Subtraction
• Multiplication
• Division
• Practice Questions

## What are Odd Numbers?

Any number which cannot be devided by two is termed as an odd number. In other words, A number of the 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 key points related to odd numbers are:

• 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 List:

There are almost 24 odd numbers from 1 to 50 while there are 50 odd numbers from 1 to 100. In case of odd numbers from 1 to 100, 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. while even number list include -6, -4, -2, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 etc. The list of odd numbers from 1 to 100 is given in th chart below.

## Types of 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.

## Properties of Odd Numbers:

There are four main properties of odd numbers which are related to their addition, subtraction, multiplication, and division. Each of these odd number properties are discussed in the following points in a detailed way along with proves and examples.

• ### 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 number be a and b.

These numbers can be written in the form where

$a = 2k_{1}+1$

and $b = 2k_{2}+1$

where $k_{1}, k_{2}\in \mathbb{Z}$

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

 Lets Work Out: Example- 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.
• ### Subtracting Two Odd Numbers

When an odd number is subtracted from an even 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 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

Proof:

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 $k_{1}, k_{2}\in \mathbb{Z}$ $a \times b = (2k_{1}+1)(2k_{2}+1)$ $a \times b = 4k_{1}k_{2}+2k_{1}+2k_{2}+1$

The above equation can be re-written as $a \times b = 2(2k_{1}k_{2}+k_{1}+k_{2})+1 = 2(x)+1$

Thus, the multiplication of two odd number results in 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 numerator, or else the number result in decimal point number.

 $\frac{Odd}{Odd} = Odd$

### Odd Number Questions for Practice:

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

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#### Practise This Question

In a hypothetical universe, all the numbers are represented using dots. Each number is represented by that many dots as the value of the number. These dots are then joined by a straight line. The number 2 when represented in this world, would only form a ___.