Even and Odd Numbers
- Even Numbers:Numbers that can be divided by 2 without leaving a remainder. These are represented as 2n, where n is an integer.
- Examples: 0, 2, 4, 6, 8, etc.
- When divided by 2, the remainder is 0.
- Odd Numbers:Numbers that leave a remainder of 1 when divided by 2. These are generally represented as 2n – 1, where n is an integer.
- Examples: 1, 3, 5, 7, 9, etc.
- When divided by 2, the remainder is 1.
Even and Odd Natural Numbers
Many objects around us can be classified as either odd or even. Some of these objects are used individually, while others are used in pairs.
Odd Numbers
- Each individual object of the same kind or a complementary pair is called odd.
- Mathematically, an odd number is defined as an integer that leaves a remainder of 1 when divided by 2.
- Odd numbers are represented as: 2k + 1 (k ∈ Z)
Even Numbers
- Even numbers are formed by combining two odd numbers.
- Mathematically, an even number is an integer divisible by 2 without any remainder.
- Even numbers are represented as: 2k (k ∈ Z)
Warning: The factorials of odd numbers are 0 and 1. (0! = 1 and 1! = 1). The factorials of all other numbers are even because, starting from 2, the factorials of all numbers contain at least one even number.
Basic Properties
Oddness and Evenness in Integers:
- Every integer is either odd or even.
- Zero (0) is an even number.
Negative Numbers:
- Negative integers can also be odd or even:
- Example: −2 is even, −11 is odd.
- If a number is odd, its negative is also odd. If a number is even, its negative is also even.
- Example: 16 and −16 are even; 47 and −47 are odd.
Operations with Odd and Even Numbers
1. Addition and Subtraction
| Operation Type | Result | Example |
|---|---|---|
| Odd + Odd | Even | 3 + 5 = 8 |
| Even + Even | Even | 4 + 6 = 10 |
| Odd + Even | Odd | 5 + 4 = 9 |
| Odd – Odd | Even | 7 − 3 = 4 |
| Even – Even | Even | 8 − 2 = 6 |
| Odd – Even | Odd | 7 − 2 = 5 |
2. Multiplication
| Operation Type | Result | Example |
|---|---|---|
| Odd × Odd | Odd | 3 × 5 = 15 |
| Even × Even | Even | 4 × 6 = 24 |
| Odd × Even | Even | 5 × 4 = 20 |
3. Exponentiation
| Operation Type | Result | Example |
|---|---|---|
| Natural power of an odd number | Odd integer | 3² = 9, 5³ = 125 |
| Natural power of an even number | Even integer | 4² = 16, 2³ = 8 |
Note: This rule applies only when the exponent is positive or zero. If the exponent is negative, the expression remains unchanged.
