Numbers are very useful in our daily life to count many things. By the divisibility rule of 2, we can classify numbers (integers) as follows: even numbers and odd numbers. Note that even numbers are completely divisible by 2 whereas odd numbers are not divisible by 2 that leaves the remainder 1. In this section, we will learn about odd numbers.

**Also Read:** **Even and odd numbers**

**Definition of Odd Numbers**

A number is called an odd number if it leaves a remainder 1 while dividing by 2.

Note that 1 is the first positive odd number.

Examples of odd numbers: 3, 5, 9, 11, 101, 107 are a few examples of odd numbers.

**Set of all Odd Numbers**

When we divide an odd number by 2, it leaves the remainder 1. So an odd number can be expressed as 2k+1 for some integer k. Thus we can write the set of all odd numbers as follows:

$\{2k+1 : k \text{ is an integer}\}$

$=2\mathbb{Z}+1,$

where $\mathbb{Z}:=$ $\{k: k \text{ is an integer}\}$ is the set of all integers. Thus we see that the general form of an odd number is 2k+1 where k is an integer.

**List of Odd Numbers from 1 to 100**

The following are the odd numbers from 1 to 100:

1, 3, 5, 7, 9,11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97 and 99.

Note that there are 50 odd numbers from 1 to 100.

**Is one an odd number?**

Note that 1=2.0+1. Thus we can write 1 as 2k+1 where k=0 is an integer. Hence by definition of an odd number 1 is an odd number.

**How to check odd numbers?**

Keep in mind that every odd number ends with either 1, 3, 5, 7, or 9. Using this fact we can check a given number is odd or not. For example, 171984231 is an odd number as it ends with 1. But the number 17190 is not an odd number as it ends with 0. It is an even number.

**Properties of Odd Numbers**

• Every even number must have the last digit either 1, 3, 5, 7, or 9.

• Odd numbers are not a multiple of 2, but even numbers are.

• Odd number + Odd number = Even number

• Odd number – Odd number = Even number

• Odd number × Odd number = Odd number

**Addition or Subtraction of Odd Numbers**

The sum of two odd numbers is always an even number.

**Proof:** Let a and b be two odd numbers. So by definition of odd numbers, we have

a=2k+1 and b=2m+1 for some integers k and m.

Adding we get that

a+b = (2k+1)+(2m+1) = 2k+2m+2 = 2(k+m+1)

So a+b is a multiple of 2. This makes a+b an even number, completing the proof.

As we have a-b=2(k-m), the difference a-b is a multiple of 2. Hence a-b is an even number. As a result, we deduce the fact: The difference between two even numbers is again an even number.

**Multiplication of Odd Numbers**

The product of two odd numbers is an odd number.

**Proof:** Let a and b be two odd numbers. So we can write

a=2k+1 and b=2m+1 for some integers k and m.

Multiplying a and b we get that

a × b = (2k+1) × (2m+1) = 4km+2k+2m+1 = 2(2km+k+m)+1

So ab can be written as ab=2t+1 where t=2km+k+m is an integer. Thus by definition of an odd number, we conclude that ab is an odd number. Hence the proof follows.