OneTaskBy Tim King · Updated
Factor Calculator
All Factors, Factor Pairs and Prime Factorisation of a Number
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How the Factor Calculator works
This factor calculator lists every factor of a whole number up to one trillion, along with the factor pairs that multiply to it and its prime factorisation, such as 360 = 2³ × 3² × 5. It also counts the factors, adds them up and says whether the number is prime, perfect, abundant or deficient.
A factor divides a number exactly, leaving no remainder. Factors come in pairs that multiply to the number, such as 4 × 9 = 36, which is why listing them only needs a search up to the square root. The prime factorisation is the number's unique fingerprint: every factor is some combination of its prime factors, and it is the quickest route to the greatest common factor and least common multiple.
How the factor calculator works
The formula for the count of factors comes from the prime factorisation: if n = p^a × q^b × …, then n has (a + 1)(b + 1)… factors. The calculator divides out primes to get the factorisation, builds every combination of them to list the factors, and pairs the smallest with the largest. A perfect number's factors, excluding itself, add up to the number.
First example. 36 = 2² × 3², so it has (2 + 1)(2 + 1) = 9 factors: 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Second example. 28 has factors 1, 2, 4, 7, 14 and 28. The first five add to 28, so it is a perfect number.
How many factors does 360 have? 360 = 2³ × 3² × 5, so (3 + 1)(2 + 1)(1 + 1) = 24 factors.
Numbers with many factors, with their prime factorisation
| Number | Prime factorisation | How many factors | Factors |
|---|---|---|---|
| 6 | 2 × 3 | 4 | 1, 2, 3, 6 |
| 12 | 2² × 3 | 6 | 1, 2, 3, 4, 6, 12 |
| 18 | 2 × 3² | 6 | 1, 2, 3, 6, 9, 18 |
| 24 | 2³ × 3 | 8 | 1, 2, 3, 4, 6, 8, 12, 24 |
| 28 | 2² × 7 | 6 | 1, 2, 4, 7, 14, 28 |
| 30 | 2 × 3 × 5 | 8 | 1, 2, 3, 5, 6, 10, 15, 30 |
| 36 | 2² × 3² | 9 | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
| 48 | 2⁴ × 3 | 10 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 |
| 60 | 2² × 3 × 5 | 12 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 |
| 72 | 2³ × 3² | 12 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 |
| 84 | 2² × 3 × 7 | 12 | 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 |
| 90 | 2 × 3² × 5 | 12 | 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 |
| 96 | 2⁵ × 3 | 12 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 |
| 100 | 2² × 5² | 9 | 1, 2, 4, 5, 10, 20, 25, 50, 100 |
| 120 | 2³ × 3 × 5 | 16 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24 … |
| 144 | 2⁴ × 3² | 15 | 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36 … |
| 180 | 2² × 3² × 5 | 18 | 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20 … |
| 240 | 2⁴ × 3 × 5 | 20 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20 … |
| 360 | 2³ × 3² × 5 | 24 | 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18 … |
| 496 | 2⁴ × 31 | 10 | 1, 2, 4, 8, 16, 31, 62, 124, 248, 496 |
| 720 | 2⁴ × 3² × 5 | 30 | 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16 … |
| 1000 | 2³ × 5³ | 16 | 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125 … |
| 5040 | 2⁴ × 3² × 5 × 7 | 60 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14 … |
| 8128 | 2⁶ × 127 | 14 | 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032 … |
Tips and pitfalls
- Square numbers have an odd count. Their middle pair is the same number twice, such as 6 × 6.
- Factors versus multiples. Factors of 12 divide into 12; multiples of 12 are 12, 24, 36 and so on.
- Use a factor tree. Split into any two factors and keep splitting until every branch ends in a prime.
- Primes have two factors. If the list shows only 1 and the number, it is prime.
FAQ
How do you find all the factors of a number?
Test each whole number from 1 up to the square root. Each one that divides exactly gives a pair. For 48, √48 ≈ 6.9, so test 1 to 6: the pairs are 1 × 48, 2 × 24, 3 × 16, 4 × 12 and 6 × 8. That gives ten factors in total.
What is prime factorisation?
It writes a number as a product of primes only, such as 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. Every whole number above 1 has exactly one prime factorisation, apart from the order. It is used to simplify fractions and roots and to find the GCF and LCM.
What are factor pairs?
Factor pairs are two numbers that multiply to give the target. The factor pairs of 24 are 1 × 24, 2 × 12, 3 × 8 and 4 × 6. A square number has one pair where both are equal, such as 5 × 5 for 25, which gives it an odd number of factors.
What is a perfect number?
A perfect number equals the sum of its factors other than itself. 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. The next are 496 and 8,128. When the sum is larger the number is abundant, such as 12, and when it is smaller it is deficient.
What is the difference between a factor and a multiple?
A factor goes into a number; a multiple is what you get by multiplying it. 3 is a factor of 12, and 12 is a multiple of 3. A number has a limited list of factors but an endless list of multiples, and it is always both a factor and a multiple of itself.
See also: all Number Theory tools, Arithmetic, Algebra and Geometry, Statistics, Trigonometry and Calculus.
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