Odd Numbers

In math, some numbers cannot be divided completely by 2. Such numbers are called odd numbers. They are often represented in the form 2k + 1, where k is any integer.
Odd numbers are not divisible by 2, so it always leaves a remainder.
 

The main difference between even and odd numbers is the divisibility rule of 2. Here are a few key differences between them:
 

To easily identify an odd number, check if the given number is not a multiple of 2. An odd number often ends in numbers such as 1, 3, 5, 7, or 9. Moreover, since odd numbers cannot be divided evenly by 2, they always leave a remainder of 1.
 

The list of odd numbers from 1 to 100 lays the foundation for learning more complex odd numbers. Listing these numbers helps identify numbers not divisible by 2. 
We can easily find the odd numbers by simply adding 2 to the previous odd number. For example: adding 2 to the first odd number gives 3, i.e., 1, 1 + 2 = 3, 3 + 2 = 5, 5 + 2 = 7, and so on. Similarly, we can list all the odd numbers from 1 to 100. The last odd number in the list of 1 to 100 is 99, as adding 2 gives 101, which is greater than 100.

On a number line, odd numbers can be represented by skipping 2 after each number. Positive odd numbers move to the right of 0 (e.g., 1, 3, 5, 7…), whereas negative odd numbers move to the left of 0 (e.g., –1, –3, –5, –7…). 

The sum of Odd Numbers from 1 to 100

To find the sum of odd numbers from 1 to 100, we will use the formula given below:
S = n/2 (first odd number + last odd number)
Here,
S = Sum of the odd numbers
n = total number of odd numbers between 1 and 100. 
Given that there are 50 odd numbers in the range of 1 to 100,  n = 50.
We now substitute these values into the formula:
S = (50 /2) (1 + 99) 
S = 25 (100)
S = 2500
Thus, all odd numbers between 1 and 100 add up to 2500.
 

Odd numbers are classified into three types. Let’s now learn how each of them differs:

Consecutive Odd Numbers

When the numbers in a sequence appear one after the other without any gaps, they are referred to as consecutive numbers. If these numbers are also odd, we call them consecutive odd numbers. For example: 1, 3, 5, 7, and 9. Each consecutive odd number is simply found by adding 2 to the previous number. Note that the difference between any two consecutive odd or even numbers will always result in 2.

Composite Odd Numbers

A number is said to be composite if it has more than two factors. Composite numbers are of two different types: composite odd numbers and composite even numbers. An odd number that has more than two factors is called a composite odd number. For example, 15 can be considered a composite odd number because it has factors 1,3, 5, and itself.

Prime Odd Numbers

Numbers that are both odd and prime are called prime odd numbers. These numbers have only two factors: 1 and themselves, and they are not divisible by 2. For example: 3, 5, 7, 11, etc. Keep in mind that all prime numbers are odd except for 2, which is considered the only even prime number.
 

We can express all the odd numbers in the form: 2k +1, where k is an integer. For example, 13 can be expressed as 2 × 6 + 1, and  –11 as 2 × (-6)+ 1. We will now look at the different properties of odd numbers.
 

We’ve now learned the paramount importance of odd numbers in math. We will now learn why they are significant in real-world situations beyond math:

• Odd numbers often create symmetry, which is utilized in seating arrangements, such as in theaters or stadiums.

• Architects use these numbers to design visually pleasing structures or compositions.

• Discounts or offers are often presented in odd amounts. For example: $49 instead of $50.

• Odd numbers are used in traffic control on certain days by using odd-numbered license plates.

• Staircases are often designed with an odd number of steps to help maintain balance