Odd Numbers

Odd numbers are the numbers that cannot be divided by 2 evenly. It cannot be divided into two separate integers evenly. If we divide an odd number by 2, then it will leave a remainder. The examples of odd numbers are 1, 3, 5, 7, etc. Odd numbers are just the opposite concept of even numbers.  

The most simple way to remember an odd number is ‘it is not a multiple of 2’.  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.

What are Odd Numbers?

Odd numbers are defined as any number which cannot be divided by two. In other words, numbers in the form of 2k+1, where k ∈ Z (i.e. integers) are called odd numbers. It should be noted that numbers or sets of integers on a number line can either be odd or even.

Odd Numbers and Even Numbers

The numbers that are not odd, are even numbers, which are divisible by 2. Odd numbers cannot be divided into two parts equally but even numbers can be divided equally. For example, 4 can be divided and paired into two parts as 2-2, but 3 cannot be divided and paired.

Learn more: Even And Odd Numbers

Facts to Remember

Odd Numbers Chart (1 to 100)

This chart consists of odd numbers from 1 to 100. You can also practise writing the odd numbers from 1 to 1000 in your notebook.

List of Odd Numbers

There are 25 odd numbers from 1 to 50 while there are 50 in between 1 and 100. In case of numbers from 1 to 1000, there are 500 odd numbers and 500 even numbers.

Odd Numbers from 101 to 200

Odd Numbers from 201 to 300

Odd Numbers from 301 to 400

Odd Numbers from 401 to 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 is 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.

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.

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.

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 results in an odd number if and only if the denominator is a factor of the numerator, or else the number result in decimal point number.

If we summarise the properties of odd numbers, we have:

Types of Odd Numbers

There are mainly two types of odd numbers:

• Consecutive odd numbers
• 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 that 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.

What is the Smallest Odd Composite Number?

The smallest odd composite number is 9. As we know, the first few odd numbers are 1, 3, 5, 7, 9, 11, and so on. And we know that the list of the first few composite numbers are 4, 6, 8, 9, 10, 12, 14, and so on. By comparing the definition of both odd numbers and composite numbers, we can conclude that 9 is the smallest odd composite number.

i.e., List of odd numbers: 1, 3, 5, 7, 9, 11, …

List of composite numbers: 4, 6, 8, 9, 10, 12, 14, …

Video Lesson on Numbers

Solved Examples on Odd Numbers

Example 1:

Find the sum of the smallest and the largest 3 digits odd numbers 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 the divisibility rule of 2).

This proves that the number is even.

Example 2:

Are the following numbers odd or even?

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

Example 3:

Fill in the blanks with suitable odd numbers:
__, 2, __, 4, __, 6, __, 8, __, 10, __, 12.
22, __, 24, __, 26, __, 28, __, 30.
100, __, 102, __, 104, __, 106, __, 108.

Solution:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
22, 23, 24, 25, 26, 27, 28, 29, 30.
100, 101, 102, 103, 104, 105, 106, 107, 108.

Example 4: How many odd numbers are there between 1 to 100?

Solution: There are 50 odd numbers and 50 even numbers between 1 to 100.
Number of odd numbers = 100/2 = 50

Odd Numbers Worksheet

1. Is 7 an even or odd number?
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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