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.
Table of Content 
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 22, but 3 cannot be divided and paired.
Learn more: Even And Odd Numbers
Facts to Remember
A few more key points 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.
Number Range  No. of Odd Numbers 
1 to 50  25 
1 to 100  50 
1 to 200  100 
1 to 300  150 
1 to 500  250 
1 to 1000  500 
1 to 10000  5000 
Odd Numbers from 101 to 200
101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 193, 195, 197, 199. 
Odd Numbers from 201 to 300
201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 249, 251, 253, 255, 257, 259, 261, 263, 265, 267, 269, 271, 273, 275, 277, 279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299. 
Odd Numbers from 301 to 400
301, 303, 305, 307, 309, 311, 313, 315, 317, 319, 321, 323, 325, 327, 329, 331, 333, 335, 337, 339, 341, 343, 345, 347, 349, 351, 353, 355, 357, 359, 361, 363, 365, 367, 369, 371, 373, 375, 377, 379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399. 
Odd Numbers from 401 to 500
401, 403, 405, 407, 409, 411, 413, 415, 417, 419, 421, 423, 425, 427, 429, 431, 433, 435, 437, 439, 441, 443, 445, 447, 449, 451, 453, 455, 457, 459, 461, 463, 465, 467, 469, 471, 473, 475, 477, 479, 481, 483, 485, 487, 489, 491, 493, 495, 497, 499. 
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.
Odd + Odd = Even 
Proof:
Let two odd numbers be a and b.
These numbers can be written in the form where
a = 2k_{1} + 1
and b = 2k_{2} + 1 where k_{1}, k_{2} ∈ Z
Adding a + b we have,
(2k_{1} + 1) + (2k_{2} + 1) = 2k_{1} + 2k_{2} + 2 = 2(k_{1} + k_{2} + 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 = 2k_{1} + 1 and b = 2k_{2} + 1 where k_{1} , k_{2} ∈ Z
Now, a × b = (2k_{1} + 1)(2k_{2} + 1)
So, a × b = 4k_{1 }k_{2} + 2k_{1} + 2k_{2} + 1
The above equation can be rewritten as:
a × b = 2(2k_{1} k_{2} + k_{1} + k_{2}) + 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.
Odd ⁄ Odd = Odd 
If we summarise the properties of odd numbers , we have:
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 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.
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?
 25
 15 + 13
 32 – 37
Solution:
 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
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
 Is 7 an even or odd number?
 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?
Related Articles
Even Numbers  CoPrime Numbers 
Prime, Composite And Even, Odd Numbers  Natural Numbers 
Whole Numbers  Number System 
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Frequently Asked Questions on Odd Numbers
What are all the Odd Numbers?
The numbers which are not divisible by 2, evenly, are called odd numbers. For Example: 3, 5, 17, 19, 21, etc.
How do you determine if a number is odd or even?
If a number is evenly divisible by 2 then it’s an even number, otherwise, it’s an odd number. We can also say when we divide a number by 2 and there is some remainder left, which is not divisible again by 2, then it’s an odd number.
For a greater number which is in ten thousands or millions, check the number at the unit place. If unit place carries an even number, that means the whole number is even else it’s an odd one.
Is zero an odd number? Why?
No, zero is not an odd number but an even number, because, when we divide 0 by 2, it gives us a quotient equal to 0 and also there is no remainder left after division. So, 0 is evenly divided by 2.
0 ÷ 2 = 0
Is 37 an odd number or even?
As we can see, the unit place of 37 consists of an odd number i.e. 7, which is not evenly divisible by 2, therefore, it’s an odd number.