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What Is a Factor in Math?
A factor of a number is a whole number that divides into it exactly, leaving no remainder. Factors always come in pairs, which is what makes listing them systematic. This guide covers how to find every factor of a number, what makes a number prime, prime factorisation, the greatest common factor, and the difference between a factor and a multiple. Thefactor calculator lists every pair for any number.
7 min read
What is a factor in math?
A factor of a number is a whole number that divides into it exactly, leaving no remainder. 3 is a factor of 24 because 24 / 3 = 8 with nothing left over. 5 is not, because 24 / 5 leaves a remainder of 4.
Factors always come in pairs. Because 3 × 8 = 24, finding 3 hands you 8 at the same time. That pairing is what makes listing them systematic rather than a guess.
How to find all the factors of a number
Work upward from 1 and record both members of each pair. Stop when the pair meets in the middle.
For 24:
- 1 × 24
- 2 × 12
- 3 × 8
- 4 × 6
Next would be 5, which does not divide, then 6 — and 6 has already appeared as the partner of 4, so the list is complete.
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24: eight of them. Every number has at least two factors, 1 and itself, and 1 is the only number whose two are the same.
Factors and products are one statement
Every multiplication can be read in both directions. 3 × 8 = 24 says 24 is the product of 3 and 8, and equally that 3 and 8 are factors of 24. Nothing changes but which end you start from.
Multiplying is the easy direction. Going backwards — given 24, find everything that produces it — is the work, and it is what the factor calculator automates. what a product is covers the forward direction.
Prime factors and prime factorisation
A prime number has exactly two factors: 1 and itself. 7 is prime because its only factors are 1 and 7. A number with more than two is composite.
Prime factorisation breaks a number into primes multiplied together. For 24: 24 = 2 × 2 × 2 × 3, usually written 2³ × 3. Every whole number above 1 has exactly one prime factorisation, which is why it is the fingerprint of a number rather than just one description of it.
Note that 1 is neither prime nor composite. It has only one factor, so it fails the definition of prime, which requires exactly two.
Common factors and the GCF
A common factor divides two numbers exactly. Compare 24 and 36:
| Number | Factors |
|---|---|
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
| Shared | 1, 2, 3, 4, 6, 12 |
They share six factors, and the largest is 12 — the greatest common factor. The GCF is what you divide by to reduce a fraction to lowest terms: 24/36 becomes 2/3 by dividing both parts by 12.
Factor vs multiple
These two get swapped constantly, and they point in opposite directions.
- A factor of 24 divides into it: 1, 2, 3, 4, 6, 8, 12, 24. There are finitely many, and none is larger than 24.
- A multiple of 24 is what 24 divides into: 24, 48, 72, 96, and so on forever. There are infinitely many, and none is smaller than 24.
Factors go down, multiples go up. A number is both a factor and a multiple of itself, which is the one place the two lists touch.
Common mistakes
- Forgetting 1 and the number itself. Both are always factors.
- Confusing factors with multiples. Factors divide in; multiples are divided into.
- Calling 1 prime. Prime means exactly two factors; 1 has one.
- Stopping the search too early and missing a pair. Work up until the pair meets in the middle.
Quick summary
- A factor divides a number exactly, with no remainder.
- Factors come in pairs; 24 has eight of them.
- A prime has exactly two factors; 1 has one, so it is not prime.
- The GCF of 24 and 36 is 12, which is what reduces 24/36 to 2/3.
The factor calculator lists every factor pair for any number.