Factor Calculator
List every positive factor of a whole number and write its prime factorization. Factors come in pairs: if d divides n, so does n/d. Walking d from 1 to √n finds them all without duplicates.
Factor Calculator tool
About the Factor Calculator
The default 84 is 2² × 3 × 7 and has 12 factors: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. If the number is prime, the factorization is the number itself and the factor list is 1 and n.
This is not factorial n!. Factorial is 84 × 83 × … × 1 on the scientific calculator. It is also not an LCM/GCF list — those tools take several numbers at once.
How the Factor Calculator works
If d divides n and d ≤ √n, record d and n/d. Sort the list. For prime factorization, divide out the smallest prime factor repeatedly until n becomes 1.
If d | n and d ≤ √n, then both d and n/d are factors
n = p₁^a₁ × p₂^a₂ × … (n > 1)
Example
84 = 2² × 3 × 7, with 12 positive factors.
How to Use the Factor Calculator
- Enter a positive whole number from 1 through 1,000,000,000,000.
- Read the prime factorization as the main result.
- Use the factor list for “all divisors” homework.
- The Prime? flag is yes only when the factorization is a single prime to the first power.
- Open the GCF or LCM calculator to combine several numbers.
Factor Calculator Examples
Composite with several primes
84 = 2² × 3 × 7. Factor count (2+1)×(1+1)×(1+1) = 12.
A prime
13 is prime. Factors: 1, 13. Factorization: 13.
A square
36 = 2² × 3². Factors include 6 twice as 36/6, recorded once: 1, 2, 3, 4, 6, 9, 12, 18, 36.
One
1 has a single factor, 1, and no prime factorization other than the number 1 itself.
How to Find Factors of a Number
Test integers from 1 up to the square root. If n ÷ d is a whole number, both d and n ÷ d are factors. Stop at the square root so you do not double-count. For 84, √84 ≈ 9.2, so you only test 1 through 9 and pick up 84, 42, 28, 21, 14, 12 as the partners of 1, 2, 3, 4, 6, 7. That is the method this factor calculator uses.
Prime factorization versus the factor list
Prime factorization is unique (the fundamental theorem of arithmetic). The factor list can be rebuilt from the exponents: each exponent runs from 0 through its maximum. 2^a × 3^b × 7^c with a ≤ 2, b ≤ 1, c ≤ 1 produces the 12 factors of 84. If you only needed “is it prime?”, the prime number checker stops at the first factor.
Factor examples
Use the table as a worksheet key. Perfect squares have an odd number of factors because one divisor is repeated at the square root. Primes always have exactly two.
| n | Prime factorization | Factor count |
|---|---|---|
| 1 | 1 | 1 |
| 13 | 13 | 2 |
| 36 | 2^2 × 3^2 | 9 |
| 84 | 2^2 × 3 × 7 | 12 |
| 100 | 2^2 × 5^2 | 9 |
| 97 | 97 | 2 |
GCF and LCM from factorizations
GCF keeps the lowest exponent of each shared prime. LCM keeps the highest exponent of every prime that appears. Factor 12 and 18 here, then confirm on the GCF and LCM pages: GCF = 2 × 3 = 6, LCM = 2² × 3² = 36. This page still factors only one integer at a time.
Frequently Asked Questions
Are negative factors included?
No. The list is positive divisors only.
Is this factorial?
No. Factorial n! is a product of a decreasing sequence. Use the scientific calculator for n! with n from 0 through 170.
How many factors can a number have?
If n = p^a × q^b × … the count is (a+1)(b+1)…. The tool lists them all for n up to 1e12.
What about 1?
1 is listed as the only factor. It is not prime.
Can I factor a decimal?
No. Whole numbers only.
Does this check primality?
It reports Prime? as yes or no. The prime number checker is the dedicated yes/no page and shows a smallest factor for composites.
Guides that use this calculator
- How to Find Factors of a Number
Factors come in pairs up to the square root. Here is the list for 84, prime factorization, and the free factor calculator — not factorial n!.
- How to Check a Prime Number
Prime means exactly two positive divisors. Here is the square-root test, why 1 is not prime, and the free prime number checker.
- How to Find LCM (Least Common Multiple)
The least common multiple is the smallest number that every input divides. Here is the GCF formula, a 12-18-24 example and the free LCM calculator.