What is a factor?
A factor of a whole number is a whole number that divides it exactly, with nothing left over. Because 3 × 8 = 24, both 3 and 8 are factors of 24, and so are 1, 2, 4, 6, 12 and 24. Every whole number greater than 1 has at least two factors, 1 and itself.
This page uses positive factors only, which is what school work asks for. Negative numbers can divide a number exactly too (−3 divides 24), but they are not listed here. A factor is not the same as a multiple: factors divide into a number, while multiples are what you get by multiplying it, so 3 is a factor of 24 and 24 is a multiple of 3. When you need the factors that two numbers share, use the GCF calculator.
How to find the factors of a number
To find all the factors of a number by hand, divide it by 1, 2, 3 and so on, and note each divisor that leaves no remainder together with its partner. You only need to test divisors up to the square root of the number, because every factor above the square root is paired with one below it.
Worked example: the factors of 48
- The square root of 48 is between 6 and 7 (6 × 6 = 36 and 7 × 7 = 49), so test the divisors 1 to 6.
- 48 ÷ 1 = 48, 48 ÷ 2 = 24, 48 ÷ 3 = 16, 48 ÷ 4 = 12 and 48 ÷ 6 = 8 all divide exactly. 48 ÷ 5 does not, because 5 × 9 = 45 leaves a remainder of 3.
- That gives the factor pairs 1 × 48, 2 × 24, 3 × 16, 4 × 12 and 6 × 8.
- List every number from the pairs in order: 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. So 48 has 10 factors.
To skip the division, type the number into the calculator above and press Calculate. It lists every factor and factor pair, counts the factors and adds them up. For a large number, the quickest route is to find its prime factorization first and then combine the primes, which is what the calculator does.
Prime factorization and the factor tree
The prime factorization of a number writes it as a product of prime numbers, and every whole number greater than 1 has exactly one such product, apart from the order of the factors. To write the prime factors, keep splitting the number into two smaller factors until every piece is prime, then group the repeated primes with exponents.
Worked example: the prime factors of 72
- 72equals2times36
- 36equals2times18
- 18equals2times9
- 9equals3times3
Blue circles are prime. Each step splits off the smallest prime until only primes are left.
Read the primes from the tree: 72 = 2 × 2 × 2 × 3 × 3, which is written 2³ × 3². Different trees for the same number, such as starting with 8 × 9, end with the same primes.
The exponents also tell you how many factors a number has. Add 1 to each exponent and multiply the results: for 72 that is (3 + 1)(2 + 1) = 12 factors. The same prime factors lead straight to the greatest common factor and the least common multiple of two numbers, so the GCF calculator and the LCM calculator are the natural next steps.
- Number of factors
- (e₁ + 1)(e₂ + 1)… where n = p₁^e₁ × p₂^e₂ × …
Factor pairs and proper factors
A factor pair is two whole numbers whose product is the number: 6 × 8 = 48, so 6 and 8 are a factor pair of 48. A number has half as many pairs as factors, rounded up. A perfect square has one pair made of the same number twice, such as 6 × 6 = 36.
The proper factors of a number are all its factors except the number itself. The proper factors of 12 are 1, 2, 3, 4 and 6. When the proper factors add up to the number, it is called a perfect number: 6 = 1 + 2 + 3. Some textbooks also leave out 1, so check which definition your course uses.
Is a number prime or composite?
A prime number has exactly two factors, 1 and itself. A composite number has more than two. The number 1 is neither. The calculator shows this in the Type line of the result, and for a prime it says so instead of giving a factorization. For example, 84 = 2² × 3 × 7 is composite, while 83 has no factors other than 1 and 83, so it is prime.
Quick checks by hand: an even number greater than 2 is composite; a number greater than 3 whose digits add up to a multiple of 3 is composite; a number greater than 5 that ends in 0 or 5 is composite. If no prime up to the square root of the number divides it, the number is prime.
Factors of common numbers
A quick reference for twelve numbers. Enter any of them in the calculator to see its factor pairs and factor tree as well.
| Number | Prime factorization | How many | All factors |
|---|---|---|---|
| 21 | 3 × 7 | 4 | 1, 3, 7, 21 |
| 27 | 3³ | 4 | 1, 3, 9, 27 |
| 32 | 2⁵ | 6 | 1, 2, 4, 8, 16, 32 |
| 33 | 3 × 11 | 4 | 1, 3, 11, 33 |
| 36 | 2² × 3² | 9 | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
| 40 | 2³ × 5 | 8 | 1, 2, 4, 5, 8, 10, 20, 40 |
| 45 | 3² × 5 | 6 | 1, 3, 5, 9, 15, 45 |
| 72 | 2³ × 3² | 12 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 |
| 78 | 2 × 3 × 13 | 8 | 1, 2, 3, 6, 13, 26, 39, 78 |
| 80 | 2⁴ × 5 | 10 | 1, 2, 4, 5, 8, 10, 16, 20, 40, 80 |
| 110 | 2 × 5 × 11 | 8 | 1, 2, 5, 10, 11, 22, 55, 110 |
| 216 | 2³ × 3³ | 16 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216 |
Swipe the table sideways to see every factor.
How this factor calculator works
The number is divided by candidate primes up to its square root (trial division) to find its prime factorization. All factors are then built from combinations of those primes, and factor pairs are the factors paired with their cofactors.
What factors are used for
- Finding all ways to arrange items into equal rows.
- Checking whether a number is prime.
- Preparing prime factors for GCF and LCM work.
Limits of this calculator
- Limited to 10¹⁵ so factoring is fast in the browser. Larger numbers can take too long with trial division.
- Positive whole numbers only.