What are the factors of a number?
Type a whole number to see every number that divides it, the factor pairs, and its prime factorization.
- Number of factors
- 12
The factors of 60 (12 in all) are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
- Factors
- 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Factor pairs
- 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10
- Prime factorization
- 2² × 3 × 5
- Prime number?
- No
- Sum of the factors
- 168
Number of factors: 12. The factors of 60 (12 in all) are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
How to calculate
Lists every factor of a whole number, its factor pairs, and its prime factorization, and says whether it is prime.
Example with the default inputs (Whole number 60): The factors of 60 (12 in all) are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
Method: Trial division: divide n by 2, 3, 5, 7, … up to √n for the prime factorization, then list every product of the prime powers.
- n is a whole number from 1 to 1,000,000,000,000 (10¹²).
- Only positive factors are listed; each negative factor −f also divides n.
- 1 is not a prime number.
Worked examples
Each example is checked against the calculator on every build.
- Whole number 60 gives Factors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, Number of factors 12, Factor pairs 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10, Prime factorization 2² × 3 × 5, Prime number? no, Sum of the factors 168.Source: hand calculation in content.mdx: 60 = 2² × 3 × 5, (2 + 1)(1 + 1)(1 + 1) = 12 factors
- Whole number 97 gives Factors 1, 97, Number of factors 2, Factor pairs 1 × 97, Prime factorization 97, Prime number? yes, Sum of the factors 98.Source: hand calculation in content.mdx: no prime up to √97 ≈ 9.8 (2, 3, 5, 7) divides 97
- Whole number 36 gives Factors 1, 2, 3, 4, 6, 9, 12, 18, 36, Number of factors 9, Factor pairs 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6, Prime factorization 2² × 3².Source: hand calculation in content.mdx: 36 = 2² × 3², a perfect square with an odd number of factors
- Whole number 1 gives Factors 1, Number of factors 1, Factor pairs 1 × 1, Prime factorization 1, Prime number? no, Sum of the factors 1.Source: hand calculation in content.mdx: 1 has one factor and is not prime
- Whole number 1,000,000,000,000 gives Number of factors 169, Prime factorization 2¹² × 5¹², Sum of the factors 2,499,694,822,171.Source: hand calculation in content.mdx: 10¹² = 2¹² × 5¹², (12 + 1)(12 + 1) = 169 factors; Python 3: (2**13 - 1) * (5**13 - 1) // 4 = 2499694822171
- Whole number 999,999,999,989 gives Number of factors 2, Prime number? yes.Source: Python 3 trial division: no divisor of 999,999,999,989 from 2 to 999,999 (the largest prime below 10¹²)
How it works
A factor of a whole number n is a whole number f that divides n with no remainder. The calculator works in two steps:
- Prime factorization by trial division. Divide n by 2 as many times as it goes, then by 3, 5, 7, 9, 11 and each odd number after that, as long as the divisor squared is no more than what is left. Each divisor that goes in is a prime factor, and the number of times it goes in is its power. If more than 1 is left at the end, that last part is a prime too. So n = p₁^e₁ × p₂^e₂ × …
- Every factor. Each factor picks a power from 0 up to eᵢ for each prime pᵢ, so the factors are all the products p₁^k₁ × p₂^k₂ × … with 0 ≤ kᵢ ≤ eᵢ, listed smallest first.
It then shows:
- Number of factors: (e₁ + 1) × (e₂ + 1) × …
- Factor pairs: each factor f up to √n with its partner n ÷ f.
- Prime number? Yes when n has exactly two factors (1 and n).
- Sum of the factors: all the factors added together, n included.
Assumptions
- n is a whole number from 1 to 1,000,000,000,000 (10¹²), so trial division needs at most about a million steps.
- Only positive factors are listed.
- 1 has one factor (1), its prime factorization is written as 1, and it is not a prime number.
Worked examples by hand
60. 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, and 5 is prime, so 60 = 2² × 3 × 5. It has (2 + 1)(1 + 1)(1 + 1) = 12 factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The pairs are 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10, and the factors add up to 168. 60 is not prime.
97. √97 is about 9.8, and none of 2, 3, 5 and 7 divides 97, so it is prime: its factors are 1, 97, which add up to 98.
36. 36 = 2² × 3², so it has 3 × 3 = 9 factors: 1, 2, 3, 4, 6, 9, 12, 18, 36. The pairs are 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6: a perfect square has one pair with the same number twice.
1. Its only factor is 1, so it has 1 factor and is not prime.
1,000,000,000,000 (10¹²). 10¹² = (2 × 5)¹² = 2¹² × 5¹², so it has 13 × 13 = 169 factors. They add up to (2¹³ − 1) × (5¹³ − 1) ÷ 4 = 8,191 × 305,175,781 = 2,499,694,822,171.
999,999,999,989. No number from 2 to 999,999 divides it, and 999,999² is more than it, so it is prime (it has 2 factors). It is the largest prime below 10¹².
Other questions people ask
What are the factors of a number?
The whole numbers that divide it with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12, because 12 ÷ each of them is a whole number. Every whole number has 1 and itself as factors.
How do I find all the factors of a number?
Try each whole number from 1 up to the square root of the number. Each one that divides it gives a factor pair. For 60, √60 ≈ 7.7, and 1, 2, 3, 4, 5 and 6 divide it, giving the pairs 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12 and 6 × 10.
What is a prime factorization?
The number written as a product of prime numbers, which is the same for every number whichever way you find it. 60 = 2 × 2 × 3 × 5, written 2² × 3 × 5. You can find it by dividing by the smallest prime that fits, again and again: 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, and 5 is prime.
How can I count the factors without listing them?
Add 1 to each power in the prime factorization and multiply. 60 = 2² × 3¹ × 5¹, so it has (2 + 1) × (1 + 1) × (1 + 1) = 12 factors.
What is a prime number?
A whole number greater than 1 whose only factors are 1 and itself, such as 2, 3, 5, 7, 11 and 97. A number with more factors is composite. 1 is neither, because it has only one factor.
Why do some numbers have an odd number of factors?
Factors come in pairs that multiply to the number, except when both numbers in a pair are the same. That happens only for perfect squares: 36 has the pair 6 × 6, so it has 9 factors, an odd number.
What about negative factors?
Each factor f also has a negative partner −f that divides the number, since (−f) × (−n ÷ f) = n. By convention, lists of factors show only the positive ones, as this calculator does.