A factor calculator lists every whole number that divides a given number exactly, along with how many factors it has. Formula: Factors = all integers i where 1 ≤ i ≤ n and n mod i = 0. 60 has 12 factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
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How to use the factor calculator
- Enter a positive whole number.
- Read the list of factors and the count.
What factors are
A factor of a whole number divides it exactly, with no remainder. The factor calculator lists every positive factor of the number you enter and counts them. Factors come in pairs that multiply to the number: for 60 the pairs are 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12 and 6 × 10.
Factor calculator worked example
The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60, which is 12 factors. You only need to test divisors up to the square root of the number, about 7.75 for 60, because every factor above it pairs with one below it.
| Number | Factors | Count |
|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 6 |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 9 |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 10 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 12 |
| 97 | 1, 97 | 2 |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 9 |
Prime factorization and counting factors
Every whole number greater than 1 is a product of primes in exactly one way. 60 = 22 × 3 × 5. To count the factors, add 1 to each exponent and multiply: (2 + 1)(1 + 1)(1 + 1) = 12. For 100 = 22 × 52, the count is 3 × 3 = 9. A number with exactly two factors, 1 and itself, is prime; 97 is prime.
Quick divisibility tests
- 2: the last digit is even.
- 3: the digits add up to a multiple of 3.
- 4: the last two digits form a multiple of 4.
- 5: the last digit is 0 or 5.
- 6: divisible by both 2 and 3.
- 9: the digits add up to a multiple of 9.
- 10: the last digit is 0.
Why factors matter
Factors are used to simplify fractions, find the greatest common factor and least common multiple, and arrange things into equal groups. Twenty-four chairs can be set out in rows of 1, 2, 3, 4, 6, 8, 12 or 24, because those are the factors of 24. Numbers with many factors, such as 12, 24, 60 and 360, have long been used for time and angles because they divide evenly in so many ways: an hour splits into halves, thirds, quarters, fifths, sixths and more.
Perfect and square numbers
A perfect square has an odd number of factors, because its square root pairs with itself: 36 has 9 factors. A perfect number equals the sum of its factors other than itself; 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. The factor calculator shows the full list, so you can check these properties yourself.
Large numbers
Finding factors of big numbers is slow, which is the basis of some encryption. The factor calculator tests divisors only up to the square root of your number, so it is instant even for numbers up to its limit of 1,000,000,000,000. Besides the list of factors, the factor calculator shows the factor pairs, the number of factors, the prime factorization and whether the number is prime. For negative numbers, the factors are the same as for the positive number, each with a plus or minus sign.
Using factors for GCF and LCM
To find the greatest common factor of two numbers, list the factors of each and pick the largest one they share. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24; the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36; the greatest common factor is 12. That lets you simplify the fraction 24/36 to 2/3 in one step. The least common multiple is then the product divided by the GCF: 24 × 36 ÷ 12 = 72.
Run each number through the factor calculator to get both lists, then compare them. For three or more numbers, find the common factors of the first two, then check which of those also divide the third.
Related converters and calculators
Related calculators:
- Prime factorization calculator: primes
- GCF calculator: greatest common factor
- LCM calculator: least common multiple
- Fraction calculator: fractions
Sources
OpenStax Prealgebra 2e, 2.4: Find multiples and factors.
Formula and method
Factors = all integers i where 1 ≤ i ≤ n and n mod i = 0.
Trial division up to √n (n up to 10^12); each divisor d below √n gives the pair n ÷ d.
Questions about this calculator
What are the factors of 60?
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
How do I count factors quickly?
Add 1 to each exponent in the prime factorization and multiply.
Is 97 prime?
Yes, its only factors are 1 and 97.
Why do perfect squares have an odd number of factors?
The square root pairs with itself.
What is a factor pair?
Two factors that multiply to the number.
Check the assumptions and units before relying on a result. Financial, tax and health estimates do not replace advice for your individual circumstances.
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