Even and Odd Numbers

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

$n = 2k$

where k is an integer

Examples:
k = 1 → 2(1) = 2
k = 5 → 2(5) = 10

Odd Number Formula

$n = 2k + 1$

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

Examples:
4 + 6 = 10 (Even)
3 + 5 = 8 (Even)
6 + 7 = 13 (Odd)

Subtraction

Examples:
10 − 4 = 6 (Even)
9 − 5 = 4 (Even)
8 − 3 = 5 (Odd)

Multiplication

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

$2 + 4 + 6 + \dots + 2n = n(n+1)$

Example:
2 + 4 + 6 + 8 = 20

Sum of first n odd numbers

$1 + 3 + 5 + \dots + (2n-1) = n^2$

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.