What are Even and Odd Numbers?
Numbers can be classified based on whether they are divisible by 2.
Even Numbers
Numbers that are divisible by 2 are called even numbers.
Examples:
0, 2, 4, 6, 8, 10, 12, 14, …
Odd Numbers
Numbers that are not divisible by 2 are called odd numbers.
Examples:
1, 3, 5, 7, 9, 11, 13, …
Definition Using Algebra
Even Number Formula
where k is an integer
Examples:
k = 1 → 2(1) = 2
k = 5 → 2(5) = 10
Odd Number Formula
where k is an integer
Examples:
k = 2 → 2(2)+1 = 5
k = 7 → 2(7)+1 = 15
How to Identify Even and Odd Numbers
Method 1: Divisibility by 2
- If number is divided by 2 and gives remainder 0 then the number is Even number
- Otherwise the number is Odd
Examples:
8 ÷ 2 = 4 → Even
9 ÷ 2 = remainder 1 → Odd
Method 2: Last Digit Rule
Even numbers end in:
0, 2, 4, 6, 8
Odd numbers end in:
1, 3, 5, 7, 9
Examples:
246 → Even
135 → Odd
Properties of Even and Odd Numbers
Addition
| Operation | Result |
|---|---|
| Even+Even | Even |
| Odd+Odd | Even |
| Even+Odd | Odd |
Examples:
4 + 6 = 10 (Even)
3 + 5 = 8 (Even)
6 + 7 = 13 (Odd)
Subtraction
| Operation | Result |
|---|---|
| Even-Even | Even |
| Odd-Odd | Even |
| Even-Odd | Odd |
| Odd-Even | Odd |
Examples:
10 − 4 = 6 (Even)
9 − 5 = 4 (Even)
8 − 3 = 5 (Odd)
Multiplication
| Operation | Result |
|---|---|
| Even × Even | Even |
| Even × Odd | Even |
| Odd × Odd | Odd |
Examples:
4 × 6 = 24 (Even)
3 × 8 = 24 (Even)
5 × 7 = 35 (Odd)
Division
Division does not always preserve even/odd type.
Examples:
8 ÷ 2 = 4 (Even)
6 ÷ 3 = 2 (Even)
9 ÷ 2 = 4.5 (not integer)
Even and Odd Powers
Even number raised to any power is always even
Example:
2² = 4
4³ = 64
Odd number raised to any power is always odd
Example:
3² = 9
5³ = 125
Sum of Even and Odd Numbers
Sum of first n even numbers
Example:
2 + 4 + 6 + 8 = 20
Sum of first n odd numbers
Example:
1 + 3 + 5 + 7 = 16 = 4²
Important observation:
Sum of first n odd numbers gives a perfect square.
Consecutive Even and Odd Numbers
Consecutive even numbers
2, 4, 6, 8, 10 …
Difference = 2
General form:
2k, 2k+2, 2k+4
Consecutive odd numbers
1, 3, 5, 7, 9 …
Difference = 2
General form:
2k+1, 2k+3, 2k+5
Prime Numbers and Even/Odd Numbers
Most prime numbers are odd.
Examples:
3, 5, 7, 11, 13
Exception:
2 is the only even prime number.
Reason:
All other even numbers are divisible by 2.