Factors of 360
Input
360
Expected result
24 factors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360. Sum 1,170; 360 = 2³ × 3² × 5
Calculators
Factor Calculator
List every factor of a whole number up to one trillion, with factor pairs, the count and sum of factors, and the prime factorization.
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About this tool
The Factor Calculator lists every positive factor of a whole number up to 1,000,000,000,000 in ascending order, together with its factor pairs, how many factors it has, their sum and its prime factorization in exponent form, such as 360 = 2³ × 3² × 5. It also says whether the number is prime, composite or neither, and whether it is perfect, abundant or deficient. To get there, it finds the prime factorization by trial division up to the square root of the number (at most about 500,000 divisions at the one-trillion limit) and then builds every factor from the prime powers. The factor count comes straight from the exponents: 360 has (3 + 1) × (2 + 1) × (1 + 1) = 24 factors in 12 pairs. A negative number is factored by its absolute value, since its factors are the same with either sign. Decimals and 0 are rejected with an explanation: factors are defined for whole numbers, and every nonzero number divides 0. The calculator runs in your browser, so nothing you type is uploaded.
Why use it
Factors, factor pairs and prime factors appear together, so one entry answers the usual homework questions.
Works up to one trillion, including large primes such as 999,999,999,989.
Explains the result with the factor-count formula and a perfect, abundant or deficient label.
Copy buttons give a clean comma-separated list or one factor pair per line.
Runs entirely in your browser with no sign-up.
How to use
- Type a whole number up to 1,000,000,000,000. Commas, spaces and e-notation are accepted, so 1,000,000 and 1e6 both work.
- Read the summary line: prime, composite or neither, with the number of factors and factor pairs.
- Scan All factors for the full list in ascending order, then Factor pairs for every a × b that equals your number.
- Check the prime factorization and the factor-count formula under it.
- Press Copy above a list to paste the factors or the pairs elsewhere.
When it helps
- Listing all factors of a number for math homework, such as the 24 factors of 360.
- Finding every rectangle with a given area, such as the ways to lay out 36 tiles in equal rows.
- Checking whether a large number is prime before using it in a puzzle or a program.
- Spotting perfect and abundant numbers in number theory exercises.
- Choosing group sizes that split a class, a team or a batch evenly.
Examples
A perfect square
Input
36
Expected result
9 factors in 5 pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6; 36 = 2² × 3²
A large prime
Input
999999999989
Expected result
Prime: its only factors are 1 and 999,999,999,989
Tips
- Only perfect squares have an odd number of factors, because one pair repeats a factor, as in 6 × 6 = 36.
- To count factors without listing them, add 1 to each exponent of the prime factorization and multiply: 72 = 2³ × 3² has (3 + 1) × (2 + 1) = 12 factors.
- Factors pair up around the square root, so testing divisors up to √n finds every pair.
- The factors two numbers share are exactly the factors of their greatest common factor, which the GCF Calculator lists.
- Large numbers can be typed with commas or spaces, or as 1e12; the calculator reads them exactly.
Frequently Asked Questions
How do I find all the factors of a number?
Divide it by every whole number from 1 up to its square root. Each divisor with no remainder gives a factor pair: for 36 the pairs are 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6, so the factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
What is the difference between factors and prime factors?
Factors are all the whole numbers that divide a number evenly. Prime factors are the primes that multiply together to make it. 12 has six factors (1, 2, 3, 4, 6 and 12), and its prime factorization is 2² × 3.
What are factor pairs?
Two factors whose product is the number. 360 has 12 pairs, from 1 × 360 to 18 × 20. A perfect square has one pair that repeats a factor, such as 6 × 6 for 36.
How many factors does a number have?
Write its prime factorization, add 1 to each exponent and multiply. 360 = 2³ × 3² × 5, so it has (3 + 1) × (2 + 1) × (1 + 1) = 24 factors.
Can I find the factors of a negative number?
Yes. Type it with a minus sign and the calculator lists the factors of its absolute value, because the factors are the same with either sign: −36 = −2 × 18 = 2 × (−18).
Why does 0 have no factor list?
Every nonzero whole number divides 0, so 0 has infinitely many factors. The calculator asks for a nonzero number instead.
Is 1 a prime number?
No. A prime has exactly two factors, 1 and itself. The number 1 has only one factor, so it is neither prime nor composite.
What are perfect, abundant and deficient numbers?
Add the proper factors, meaning every factor except the number itself. If the sum equals the number, it is perfect: 1 + 2 + 4 + 7 + 14 = 28. If the sum is larger, it is abundant, like 12 (sum 16). If smaller, it is deficient, like every prime.
What is the largest number I can factor here?
1,000,000,000,000 (one trillion). The slowest case at that size, a large prime such as 999,999,999,989, needs at most about 500,000 trial divisions, which a browser finishes in a fraction of a second.
Does this calculator factor polynomials?
No, it factors whole numbers only. To solve a quadratic such as x² − 5x + 6 = 0, whose roots are 2 and 3, use the Quadratic Formula Calculator.
Explore the category
Glossary
- Factor
- A whole number that divides another with no remainder; 4 is a factor of 12.
- Factor pair
- Two factors that multiply to the number, such as 3 × 4 = 12.
- Prime number
- A whole number greater than 1 whose only factors are 1 and itself.
- Composite number
- A whole number greater than 1 that has a factor other than 1 and itself.
- Prime factorization
- A number written as a product of primes, such as 360 = 2³ × 3² × 5.
- Proper factor
- Any factor of a number except the number itself.
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