Factors and Multiples: What Is the Difference?

Q: How do I find all the factors of a number quickly?

Test each whole number from 1 up to the square root of the number. Every time one divides exactly, write it down with its partner. When you pass the square root you have them all.

Factors divide in; multiples build up. This article shows both as rectangles of dots, explains why factors come in pairs, and gives a quick way to find every factor of a number.

4 min read Supports 3 lessons

Factors and multiples are two views of the same multiplication. Look at $3 \times 4 = 12$ . From the point of view of $12$ , the numbers $3$ and $4$ are its factors. From the point of view of $3$ and $4$ , the number $12$ is a multiple of each.

Twelve dots in 3 rows of 4: so 3 and 4 are factors of 12.

Factors: what divides in

A factor of $n$ is a whole number that divides $n$ with nothing left over. If you can arrange $n$ dots into a complete rectangle with $a$ rows, then $a$ is a factor.

Twelve dots can be arranged as $1 \times 12$ , $2 \times 6$ and $3 \times 4$ . So the factors of $12$ are $1, 2, 3, 4, 6$ and $12$ . Try $5$ rows and you are left with two dots over: $5$ is not a factor of $12$ .

Key idea

Every number has $1$ and itself as factors. A number with no other factors is called a prime.

Why factors come in pairs

Every rectangle has two sides. If $3$ is a factor of $12$ , then so is $12 \div 3 = 4$ . Factors always arrive in factor pairs that multiply to the number.

$36 = 1 \times 36$
Start at 1. Its partner is the number itself.

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Tip

You only need to test numbers up to the square root. Beyond it, the partners start repeating pairs you have already found.

Multiples: what builds up

The multiples of a number are its times table, carried on forever: the multiples of $4$ are $4, 8, 12, 16, 20, \dots$ A number has only a handful of factors but infinitely many multiples.

Multiples of 4 are the landing points of equal jumps.

Telling them apart

A useful check: factors are never bigger than the number; multiples are never smaller (for positive whole numbers). The factors of $12$ live between $1$ and $12$ . The multiples of $12$ start at $12$ and go up.

Worked example

Is $7$ a factor of $91$ ? Is $91$ a multiple of $7$ ?

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Divide: $91 \div 7 = 13$ with no remainder.

Watch out

Watch the wording. " $4$ is a factor of $12$ " and " $12$ is a multiple of $4$ " are both true. " $12$ is a factor of $4$ " is not.

Common questions

How do I find all the factors of a number quickly?: Test each whole number from $1$ up to the square root of the number. Every time one divides exactly, write it down with its partner. When you pass the square root you have them all.
Is a number a factor and a multiple of itself?: Yes. Every number divides itself exactly, and it is itself times $1$ .

Practise factors and multiples

The Factors & Multiples lesson has worked examples and unlimited practice that adapts to you.

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Common questions

How do I find all the factors of a number quickly?

Test each whole number from $1$ up to the square root of the number. Every time one divides exactly, write it down with its partner. When you pass the square root you have them all.

Is a number a factor and a multiple of itself?

Yes. Every number divides itself exactly, and it is itself times $1$ .