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.
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|
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|
|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 1: Find the sum of the smallest and the largest 3 digit odd number and also prove that it is divisible by 2.
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?
- 15 + 13
- 32 – 37
- 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?
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