Multiple — Definition, Formula & Examples

A multiple of a number is the result you get when you multiply that number by any whole number. For example, the multiples of 4 are 4, 8, 12, 16, 20, and so on.

An integer $b$ is a multiple of an integer $a$ if there exists an integer $k$ such that $b = a \times k$ . Every nonzero integer has infinitely many multiples.

Key Formula

\text{Multiple} = a \times k

Where:

$a$ = The base number whose multiples you are finding
$k$ = Any whole number (1, 2, 3, ...)

How It Works

To find multiples of a number, multiply it by 1, then by 2, then by 3, and keep going. The list never ends. For instance, the first five multiples of 6 are

6, 12, 18, 24, 30

. Zero is technically a multiple of every number because

a \times 0 = 0

, but when listing multiples in school, you usually start with the number itself.

Worked Example

Problem: List the first six multiples of 7 and determine whether 42 is a multiple of 7.

List multiples: Multiply 7 by each whole number from 1 to 6.

7, 14, 21, 28, 35, 42

Check 42: Divide 42 by 7 to see if the result is a whole number.

42 \div 7 = 6

Conclude: Since 6 is a whole number, 42 is indeed a multiple of 7.

Answer: The first six multiples of 7 are 7, 14, 21, 28, 35, and 42. Yes, 42 is a multiple of 7.

Why It Matters

Multiples come up whenever you add fractions with different denominators — you need a common multiple to create a shared denominator. They also appear in everyday tasks like figuring out scheduling patterns or counting equal groups.

Common Mistakes

Mistake: Confusing multiples with factors. A student might say 3 is a multiple of 12.

Correction: Multiples are always equal to or larger than the original number (when using positive whole numbers). 3 is a factor of 12, while 12 is a multiple of 3.