Odd Numbers

Odd numbers are the numbers that cannot be divided into two separate groups evenly. Here, all the concepts related to it like definition, examples, properties, types, etc. are covered. The key concepts that are covered here include the following.

Table of Content

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.

Also Check:

Odd Numbers Chart

Odd Numbers chart

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. Each of these properties are discussed in the following points in a detailed way.

  • 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 = 2k1 + 1

and b = 2k2 + 1 where k1, k2 ∈ Z

Adding a + b we have,

(2k1 + 1) + (2k2 + 1) = 2k1 + 2k2 + 2 = 2(k1 + k2 + 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 = 2k1 + 1 and b = 2k2 + 1 where k1 , k2 ∈ Z

Now, a × b = (2k1 + 1)(2k2 + 1)

So, a × b = 4k1 k2 + 2k1 + 2k2 + 1

The above equation can be re-written as:

a × b = 2(2k1 k2 + k1 + k2) + 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?

  1. 25
  2. 15 + 13
  3. 32 – 37

Solution:

  1. 25 is not divisible by 2, so odd number.
  2. 15 + 13 = 28, divisible by 2, not an odd number
  3. 32 – 37 = -5, is an odd number

Odd Number Worksheet 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?

Keep visiting BYJU’S to get more such maths lessons explained in an easy way. Also, register at BYJU’S to get access to various video lessons on different maths topics to learn in a more engaging and effective way.


Practise This Question

There are 45 birds out of 150 in a zoo which are migratory. 35 of the total birds are of  Indian origin. Which kind of birds are more in number in the zoo?