1. Introduction to Odd Numbers
Odd numbers are the numbers that cannot be divided into two equal parts. When you divide an odd number by 2, there is always a remainder of 1.
Odd numbers appear in many real-life situations, especially when something cannot be paired up completely.
2. Definition of Odd Numbers
An odd number is any integer that is not divisible by 2.
In simple words: Odd numbers are the numbers that leave a remainder of 1 when divided by 2.
Examples:
- \( 3 \div 2 = 1\) remainder 1 → odd
- \( 9 \div 2 = 4\) remainder 1 → odd
- \( -5 \div 2 = -2\) remainder 1 → odd
3. Identifying Odd Numbers
You can identify odd numbers by their last digit.
3.1. Last Digit Rule
If a number ends in 1, 3, 5, 7, or 9, it is an odd number.
Examples:
- 11 → ends in 1 → odd
- 53 → ends in 3 → odd
- 105 → ends in 5 → odd
3.2. Using Division
If dividing a number by 2 gives a decimal or remainder of 1, it is odd.
Examples:
- \( 15 \div 2 = 7.5 \) → odd
- \( 7 \div 2 = 3.5 \) → odd
4. Properties of Odd Numbers
Odd numbers follow interesting patterns when used in arithmetic.
4.1. Odd + Odd
The sum of two odd numbers is always even.
Example:
- \( 5 + 7 = 12 \)
- \( 9 + 11 = 20 \)
4.2. Odd + Even
Adding an odd number to an even number always gives an odd number.
- \( 3 + 4 = 7 \)
- \( 11 + 2 = 13 \)
4.3. Odd × Odd
The product of two odd numbers is always odd.
- \( 3 \times 5 = 15 \)
- \( 7 \times 9 = 63 \)
4.4. Odd × Even
Multiplying an odd number by an even number always gives an even number.
- \( 5 \times 2 = 10 \)
- \( 9 \times 4 = 36 \)
5. Examples of Odd Numbers
- -9
- -3
- 1
- 5
- 17
- 101
All these numbers end in 1, 3, 5, 7, or 9 and therefore are odd numbers.
6. Odd Numbers in Real Life
Odd numbers show up around us more often than we notice.
6.1. Uneven Grouping
If you can't divide items into perfect pairs, the total is odd. For example, 7 chocolates cannot be paired evenly.
6.2. House Numbers
In many streets, odd numbered houses are on one side: 1, 3, 5, 7...
6.3. Sports
Some tournaments require an odd number of rounds, such as best-of-3 or best-of-5 matches.
7. Difference Between Odd and Even Numbers
Odd numbers cannot be divided equally into two groups, while even numbers can.
7.1. Comparison Table
| Odd Numbers | Even Numbers |
|---|---|
| Not divisible by 2 | Divisible by 2 |
| End in 1, 3, 5, 7, 9 | End in 0, 2, 4, 6, 8 |
| Examples: 1, 3, 9 | Examples: 2, 8, 14 |
8. Practice Questions
- Is 37 an odd number?
- Write the next five odd numbers after 21.
- Is −9 an odd number?
- Which of these are odd: 42, 55, 78, 91?
- Is 0 an odd number?
9. Summary
Odd numbers are integers that are not divisible by 2. They end in 1, 3, 5, 7, or 9 and follow predictable patterns when added or multiplied. Odd numbers often appear in real-life situations where items cannot be paired evenly.