What is a factor?
A factor of a number is a whole number that divides it with no remainder. Because 6 × 5 = 30, both 6 and 5 are factors of 30, and so are 1, 2, 3, 10, 15, and 30. Every whole number greater than 1 has at least two factors, 1 and itself. A number with exactly those two factors is a prime number.
This factor calculator lists every factor of a positive whole number up to 10 trillion, along with its factor pairs, its prime factorization, and a factor tree.
How to find factors of a number
- Start with 1 and divide the number by each whole number in turn.
- When a division leaves no remainder, both the divisor and the quotient are factors. For 120, 120 ÷ 8 = 15 gives the factors 8 and 15.
- Stop once the divisor is larger than the square root of the number. Any factor beyond that point is already the quotient of a smaller one. For 120, the square root is about 10.95, so testing 1 through 10 finds all 16 factors: the factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.
The calculator uses this same method, then lists the factors of a number from smallest to largest.
Factor pairs
Factor pairs are two factors that multiply to the number. Each divisor up to the square root gives one pair: the factor pairs of 120 are 1 × 120, 2 × 60, 3 × 40, 4 × 30, 5 × 24, 6 × 20, 8 × 15, and 10 × 12. A perfect square has one pair made of the same number twice, such as 6 × 6 for 36, which is why it has an odd number of factors.
Prime factorization and the factor tree
Prime factorization writes a number as a product of prime numbers only. Every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors. For example, 120 = 2 × 2 × 2 × 3 × 5, or 2³ × 3 × 5 in exponent form.
A factor tree shows how to get there. Split the number into its smallest prime factor and the quotient, then keep splitting the quotient until it is prime: 120 → 2 × 60, 60 → 2 × 30, 30 → 2 × 15, and 15 → 3 × 5. The primes at the ends of the branches are the prime factors.
List of factors for common numbers
| Number | Factors | Prime factorization |
|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 2² × 3 |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 2³ × 3 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 2² × 3² |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 2⁴ × 3 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 2² × 3 × 5 |
| 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 | 2³ × 3² |
| 120 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 | 2³ × 3 × 5 |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 2² × 5² |
Frequently asked questions
What are the factors of a number?
They are all the whole numbers that divide it evenly. The factors of a number always include 1 and the number itself, and every other factor lies between them.
Is 1 a prime number?
No. A prime number has exactly two different factors, and 1 has only one. That is also why 1 has no prime factorization.
What is the difference between factors and multiples?
Factors divide into a number, and multiples are what you get by multiplying it. 4 is a factor of 12, while 12, 24, and 36 are multiples of 4. A number has a limited list of factors but endless multiples.
Why does the calculator stop at 10 trillion?
No known method can factor very large numbers quickly, and the time trial division takes grows with the square root of the number. Up to 10 trillion, the calculator returns results almost instantly in your browser.
What's changed
- Launched.
References
- Wolfram MathWorld, Divisor
a factor is a number that divides another, counting positive divisors only
- Wolfram MathWorld, Prime Factorization
writing a positive integer as a product of primes, each raised to its order