Odd Numbers: Definition, Types, Examples and Properties
Odd numbers are integers not divisible by two, leaving a remainder of one divided by two. Study notes on odd numbers, including its definition, types, examples and properties, are below.
Khushboo Goyal27 Oct, 2025
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Odd Numbers: Odd numbers play a key role in mathematics as they are integers that cannot be divided by two without a remainder. These numbers have special properties that set them apart from even numbers. They are important in various mathematical concepts and applications.
The odd numbers – definition and examples help students understand numbers that cannot be divided evenly by 2. When we ask what are odd numbers, we learn they always leave a remainder of 1 when divided by 2.
Learning about odd numbers helps build a strong foundation for understanding other mathematical ideas and solving problems. The unique patterns and characteristics of odd numbers are detailed below. For further learning, odd numbers - definition and examples, study notes on odd numbers are provided below.
Lets us learn odd numbers - definition and examples. Odd numbers are whole numbers that cannot be divided evenly by 2. When divided by 2, they leave a remainder of 1. Odd numbers are represented as 2n+1, where n is any integer.
The definition of odd number states that they end with digits 1, 3, 5, 7, or 9. Looking at odd numbers 1 to 100, such as 1, 3, 5, 7, and 9, gives a clear odd number meaning in maths.
Examples of odd numbers include 1, 3, 5, 7, and 9. These numbers play an important role in mathematics and have special characteristics that set them apart from even numbers.
What are Odd Numbers?
Odd numbers are numbers that cannot be divided evenly by 2. When you divide an odd number by 2, there is always a remainder of 1. In simple terms, these are numbers that end with 1, 3, 5, 7, or 9. For example, 1, 3, 5, 7, 9, 11, and 13 are all odd numbers.
The odd number meaning refers to numbers that are not exactly divisible by 2. These numbers always leave a remainder of 1 when divided by 2. In other words, odd numbers represent quantities that can’t be paired evenly. Examples include 1, 3, 5, 7, 9, and so on, ending with 1, 3, 5, 7, or 9.
According to the odd number definition, these are numbers that differ from even numbers. The sequence of consecutive odd numbers, like 1, 3, 5, 7, and 9, continues endlessly. Understanding consecutive odd numbers also helps in solving arithmetic and number pattern problems. Odd numbers are integers that cannot be divided evenly by 2.
When divided by 2, they leave a remainder of 1. Odd numbers can be further classified based on their properties. Two important types of odd numbers are Consecutive Odd Numbers and Composite Odd Numbers .
1. Consecutive Odd Numbers
Consecutive odd numbers are a series of odd numbers where each number is 2 units greater than the previous one. The difference between any two consecutive odd numbers is always 2.
Examples: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...
The general pattern of consecutive odd numbers can be written as:
First odd number: 2n+1
Second odd number: 2n+3
Third odd number: 2n+5
And so on...
Where n is a whole number.
Properties:
The sum of any two consecutive odd numbers is always an even number.
The sum of any set of consecutive odd numbers is divisible by the number of terms.
Example: For the first three consecutive odd numbers: 1, 3, and 5. The sum is 1+3+5=91 + 3 + 5 = 91+3+5=9, which is divisible by 3 (the number of terms).
2. Composite Odd Numbers
Composite odd numbers are odd numbers that are not prime. A composite number is a positive integer greater than 1 that has more than two divisors. In other words, it can be divided by numbers other than 1 and itself. Composite odd numbers are odd numbers that have divisors other than just 1 and themselves.
Examples: 9, 15, 21, 25, 27, 33, 35, ...
For example:
9 is divisible by 3.
15 is divisible by 3 and 5.
21 is divisible by 3 and 7.
These numbers are not prime because they have divisors other than 1 and the number itself.
Properties:
All composite odd numbers are greater than 1.
A composite odd number is always the product of two or more odd numbers.
These numbers are not prime, but they are still odd.
Example: Consider the number 15. It is divisible by 1, 3, 5, and 15. Since it has more divisors than just 1 and 15, it is a composite odd number.
Odd Numbers Examples
Question: Find the sum of the smallest and largest 3-digit odd numbers. Prove that the sum is divisible by 2.
Answer:
The smallest 3-digit odd number is 101. The largest 3-digit odd number is 999. The sum is 101 + 999 = 1100. To prove that 1100 is divisible by 2, we must check if it is an even number. An even number can be evenly divided by 2.
Since 1100 ÷ 2 = 550, which is a whole number, we can conclude that 1100 is divisible by 2.
Odd numbers are integers that are not divisible by 2. They always end with 1, 3, 5, 7, or 9 in the ones place. Below is a table of odd numbers 1 to 100.
Odd Numbers 1 to 100
Odd Numbers 1 to 50
Odd Numbers 50 to 100
1
51
3
53
5
55
7
57
9
59
11
61
13
63
15
65
17
67
19
69
21
71
23
73
25
75
27
77
29
79
31
81
33
83
35
85
37
87
39
89
41
91
43
93
45
95
47
97
49
99
Properties of Odd Numbers
Odd numbers are numbers that cannot be evenly divided by 2. They always leave a remainder of 1 when divided by 2. Odd numbers 1 to 100 have unique characteristics that set them apart from even numbers, and these properties are important for various areas of mathematics. Below are some key properties of odd numbers:
1. Divisibility : Odd numbers cannot be divided by 2 without leaving a remainder. This means that when you divide an odd number by 2, there will always be a remainder of 1. Example : 7 divided by 2 gives 3 with a remainder of 1, making 7 an odd number. 2. Sum of Two Odd Numbers : The sum of two odd numbers is always even. This is because adding two odd numbers together results in a number that is divisible by 2. Example : 3 + 5 = 8 (even). 3. Difference of Two Odd Numbers : When you subtract one odd number from another odd number, the result will always be even. This is because both odd numbers leave a remainder of 1 when divided by 2, and subtracting them cancels out the remainder. Example : 9 − 5 = 4, which is even. 4. Product of Two Odd Numbers : The product of two odd numbers is always odd. This means that multiplying two odd numbers together will result in a number that is also odd. Example : 3 × 5 = 15, which is odd.5. Square of an Odd Number : The square of an odd number is always odd. This means if you multiply an odd number by itself, the result will always be odd. Example : 7 2 = 49, which is odd. 6. Sum of an Odd and an Even Number : When you add an odd number and an even number together, the result is always odd. This is because adding an odd number (which leaves a remainder of 1 when divided by 2) to an even number (which is divisible by 2) will give a remainder of 1. Example : 5 + 4 = 9 , which is odd.7. Sequence of Odd Numbers : Odd numbers follow a regular pattern, starting from 1 and increasing by 2 each time. The sequence of odd numbers is 1, 3, 5, 7, 9, 11, and so on.
These properties of odd numbers are fundamental in understanding how numbers behave in arithmetic and other areas of mathematics. They help solve equations, recognize patterns, and perform calculations more efficiently.
Odd numbers are whole numbers that cannot be evenly divided by 2. They always leave a remainder of 1 when divided by 2. Identifying odd numbers is simple once you understand their key characteristics. Here are some easy methods to recognize odd numbers:
1. Definition of Odd and Even Numbers
Odd Numbers : Odd numbers are integers that cannot be divided by 2 without leaving a remainder. They always leave a remainder of 1 when divided by 2.
Even Numbers : Even numbers are integers that can be evenly divided by 2, meaning they leave no remainder.
Example of Odd Numbers : 1, 3, 5, 7, 9, 11, 13, etc. Example of Even Numbers : 2, 4, 6, 8, 10, 12, etc.
2. Mathematical Characterization
An odd number can be written as: n = 2k + 1
Where:
nnn is the odd number
kkk is any integer (positive, negative, or zero) This equation shows that any odd number is always 1 more than a multiple of 2.
3. Using the Last Digit
One of the easiest ways to identify an odd number is by looking at the last digit:
If the last digit of a number is 1, 3, 5, 7, or 9, then the number is odd.
Example :
23 ends in 3, so it is odd.
56 ends in 6, so it is even.
4. Divisibility Rule for 2
Odd numbers are not divisible by 2 without a remainder. To check if a number is odd, divide it by 2:
Another way to check if a number is odd is using the modulus operator (%). If the result of a number modulo 2 equals 1, the number is odd.
Example :
13 % 2 = 1 → Odd
24 % 2 = 0 → Even
6. Common Odd Numbers
Odd numbers follow a pattern and include:
Positive Odd Numbers : 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, and so on.
Negative Odd Numbers : -1, -3, -5, -7, -9, -11, -13, -15, -17, -19, and so on.
7. Special Case of Zero
Zero is not an odd number. It is classified as an even number because it is divisible by 2 without a remainder (0 ÷ 2 = 0).
Identifying odd numbers is easy once you know their key characteristics. You can quickly determine whether a number is odd by checking the last digit, using the divisibility rule, or applying the modulus operation. Odd numbers are important in mathematics and everyday applications, making them a fundamental concept in number theory.
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