# 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 2-2, but 3 cannot be divided and paired.

### Facts to Remember

 A few more key points to remember: An odd number is an integer which is not a multiple of 2 If these numbers are divided by 2, there will be a remainder left In the number line, 1 is the first positive odd number

## 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.

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 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?

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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## 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.