Odd Numbers

Odd numbers are numbers that are not divisible by 2 and always leave a remainder of 1 when divided by 2. They end with 1, 3, 5, 7, or 9.

Examples: 1, 3, 5, 7, 9, 11, 13, …

Key Features of Odd Numbers:

• The sequence of odd numbers starts from 1 and continues infinitely.
• They are used in number patterns, arithmetic operations, and problem-solving.
• Every odd number can be expressed in the form 2n + 1, where n is a whole number.
  For example, if n = 2, then the odd number is 2(2) + 1 = 5.

How to Identify Odd Numbers?

Numbers ending with 1, 3, 5, 7, and 9 are Odd numbers, as only numbers ending with 0, 2, 4, 6, and 8 are divisible by 2. Also if on dividing the number with 2 if the remainder is one then the number is an odd number.

Example: Which of the following is an odd number? 1123, 3214, 12452, 34824, and 98354

Solution:

From the given number 1123 is an odd number because, on dividing with 2 it gives the remainder as 1.

Even and Odd Numbers

There are some differences between even and odd numbers, as follows:

Sum of Odd Numbers from 1 to 100

The sum of all odd numbers from 1 to 100 can be calculated using the formula S = n/2(first odd number + last odd number), where n is the total count of odd numbers within the range. As there are 50 odd numbers (n = 50) between 1 and 100, we can substitute these values into the formula:

S = \frac{50}{2}(1 + 99)

This simplifies to:

S = 25 \times 100

Resulting in:

S = 2500

Therefore, the sum of all odd numbers from 1 to 100 is 2500.

Odd Number Formula

An odd number is a number that is not divisible by 2. Odd numbers can be represented in a general form using a simple formula.

The formula for an odd number is

2n + 1

where, n is any integer.

When different integer values are substituted for n, the expression 2n + 1 always produces an odd number. This formula helps in identifying, generating, and proving properties related to odd numbers in mathematics.

Properties of Odd Numbers

All the Odd numbers can be represented as 2k + 1, where all k belongs to integers. For example, 13 can be written as 2 × 6 + 1, -11 can be written as 2 × (-6) + 1, and 21 can be written as 2 × 10 + 1, etc.

There are various property of odd numbers explained in the table below,

Types of Odd Numbers

Various types of Odd Numbers are as follows,

1. Consecutive Odd Numbers

For any number to be consecutive, they need to follow each other in order, and if numbers are consecutive as well as Odd in nature then those are called consecutive odd numbers. Examples of consecutive odd numbers include 1, 3, 5, 7, and 9 (the first five consecutive odd natural numbers), and 11, 13, 15, 17, and 19. If we have an odd number a, we can determine the next consecutive odd number by adding 2 to it, i.e., a+2. It is important to note that the difference between any two consecutive odd or even numbers is always 2.

2. Composite Odd Numbers

Positive integers that have factors other than 1 and themselves are called composite numbers. For a number to be considered a composite odd number, a number must be both odd and composite. For instance, 9 is a composite odd number because it is divisible by 3, and when divided by 2, it gives a remainder of 1. Other examples of composite odd numbers include 15, 27, 35, 65, and so on.

Note: All Prime Numbers are Odd Numbers except for 2 which is Even Number.

Odd Number on Number Line

A number line is a line in which numbers are marked and is used to marks the position of various numbers and perform all sort of mathematical operations such as addition, subtraction, and others.

Odd numbers are easily represented on the number line. They are represented by skipping one number and marking the other number starting from any odd number.

Related Articles:

• Natural Numbers
• Real Numbers
• Imaginary Numbers
• Number System

Solved Examples of Odd Numbers 1 to 100

Example 1: How many odd numbers are between 1 and 150 (including 1 and 150)?

Solution:

Every other number is an odd number thus half of the all the numbers are odd.

So, between 1 and 150 (including 1 and 150), there are 150 numbers, 

Thus, half of 150 number are odd. 

There are 75 odd numbers between 1 and 150.

Example 2: Find the units digit of 3²⁰¹.

Solution:

The units digit of any power of 3 is cyclical and follows a pattern. The pattern for 3 is 3, 9, 7, 1. 

Therefore, the units digit of 3²⁰¹ is the same as the units digit of 3^x where x is the remainder when 201 is divided by 4.

and the remainder when 201 is divided by 4 is 1, so the units digit of 3²⁰¹ is the same as the units digit of 3¹, which is 3.

Therefore, the unit digit of 3²⁰¹ is 3.

Example 3: Find the product of all odd numbers between 1 to 9.

Solution:

Odd numbers between 1 to 9 are 1, 3, 5, 7, 9.

Product of all odd numbers between 1 to 9 are

= 1 × 3 × 5 × 7 × 9

= 945

Example 4: Determine whether the following numbers are even or odd,

• 73
• 2 + 4 + 6 + 8
• 99 - 67

Solution:

73 is not divisible by 2, so it is an odd number

Sum of first four even numbers is 2 + 4 + 6 + 8 = 20. Since 20 is divisible by 2, it is not an odd number, so it is an even number

99 - 67 = 32. Since 32 is divisible by 2, it is not an odd number, so it is an even number

Thus, only 73 is Odd Number

Example 5: Find the sum odd numbers from 10 to 20.

Solution:

Odd Numbers from 10 to 20 are 11, 13, 15, 17

Sum = 11 + 13 + 15 + 17

Sum = 56

Thus, the sum of odd numbers from 10 to 20 is 56.

Example 6: Find the difference of 27 and 13

Solution:

Difference of 27 and 13

= 27 - 13

= 14

Practice Questions on Odd Numbers 1 to 100

Question 1: Find the sum of odd numbers from 20 to 40

Question 2: Check wether they are odd numbers or not, 78, 23, 46, 91.

Question 3: Find the product of 13 and 21.

Question 4: How many odd numbers are from 50 to 100?