Odd Number — Definition, Examples & Properties

Odd Number

An integer that is not a multiple of 2. The odd numbers are { . . . , –3, –1, 1, 3, 5, . . . }.

See also

Key Formula

n = 2k + 1

Where:

$n$ = any odd number
$k$ = any integer (…, −2, −1, 0, 1, 2, …)

Worked Example

Problem: Determine whether 47 is odd or even.

Step 1: Divide the number by 2 and check the remainder.

47 \div 2 = 23 \text{ remainder } 1

Step 2: Since the remainder is 1, the number is not evenly divisible by 2.

Step 3: Equivalently, write it in the form

2k+1

: set

k = 23

, giving

2(23)+1 = 47

. This confirms 47 is odd.

47 = 2(23) + 1

Answer: 47 is an odd number.

Why It Matters

Odd and even classification is fundamental in number theory and algebra. Many divisibility rules, proofs, and pattern-recognition problems depend on knowing whether a number is odd or even. For instance, the sum of two odd numbers is always even, a fact used frequently in mathematical reasoning.

Common Mistakes

Mistake: Thinking that odd numbers must be positive (e.g., forgetting that

-7

is odd).

Correction: Odd numbers include negative integers. Any integer of the form

2k + 1

is odd, regardless of sign. For example,

-7 = 2(-4) + 1

Related Terms

Even Number — Integers divisible by 2; the complement of odd
Integers — The set from which odd numbers are drawn
Divisibility — Odd means not divisible by 2
Parity — The property of being odd or even