Factor Calculator
Type a whole number to get all of its factors, the factor pairs, its prime factorization and the trial-division steps, instantly.
Any whole number from 1 to 1,000,000,000,000.
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
- Prime factorization
- 2³ × 3² × 5
- Prime?
- No
- Sum of factors
- 1,170
- Sum of proper factors
- 810
- Factor pairs
- 12
- Perfect square?
- No
| Factor | × Partner | = Product |
|---|---|---|
| 1 | 360 | 360 |
| 2 | 180 | 360 |
| 3 | 120 | 360 |
| 4 | 90 | 360 |
| 5 | 72 | 360 |
| 6 | 60 | 360 |
| 8 | 45 | 360 |
| 9 | 40 | 360 |
| 10 | 36 | 360 |
| 12 | 30 | 360 |
| 15 | 24 | 360 |
| 18 | 20 | 360 |
How these factors were found (trial division)
- Every factor pairs with 360 ÷ (that factor), so you only need to test whole numbers from 1 up to √360 ≈ 18.97, i.e. 1 through 18. That search finds 12 pairs, giving 24 factors in total.
- Divide by 2 as many times as it goes: 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45. So 2³ is part of the factorization and 45 is left.
- Divide by 3 as many times as it goes: 45 ÷ 3 = 15, 15 ÷ 3 = 5. So 3² is part of the factorization and 5 is left.
- What is left, 5, is itself prime, so the division stops.
- Prime factorization: 360 = 2³ × 3² × 5.
- Number of factors = add 1 to each exponent and multiply: (3 + 1) × (2 + 1) × (1 + 1) = 24.
- Sum of factors = (1 + 2 + 4 + 8) × (1 + 3 + 9) × (1 + 5) = 1,170.
How this factor calculator finds every factor
A factor of a number is any whole number that divides it with nothing left over. The factor calculator uses the same method you would use on paper, just faster: it tests each whole number starting at 1 and checks whether the division comes out even. The trick that keeps it quick is that factors come in pairs. If 4 divides 360, then 360 ÷ 4 = 90 is also a factor, so you only have to test numbers up to the square root of your number (about 18.97 for 360) and every factor above that falls out of a pair.
The prime factorization is found by repeated division: divide by 2 as many times as it goes, then by 3, then by 5, and so on, until what is left is 1. Open "How these factors were found" under the result to see each division written out for your number.
Worked example: the factors of 360
Testing 1 through 18 against 360 gives twelve pairs: 1 × 360, 2 × 180, 3 × 120, 4 × 90, 5 × 72, 6 × 60, 8 × 45, 9 × 40, 10 × 36, 12 × 30, 15 × 24 and 18 × 20. Listed in order, that is 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.
Dividing out the primes goes 360 → 180 → 90 → 45 (three 2s), 45 → 15 → 5 (two 3s), and 5 is prime, so 360 = 2³ × 3² × 5. A neat check: add 1 to each exponent and multiply, (3 + 1)(2 + 1)(1 + 1) = 24, which matches the count above. That is why 360 was such a popular number for ancient astronomers and why there are 360 degrees in a circle: it splits evenly so many ways.
Factors, prime factors and prime numbers
Students mix these up constantly. Factors are all the numbers that divide evenly. Prime factors are only the factors that happen to be prime (for 360: 2, 3 and 5). The prime factorization is those primes with exponents, multiplied back together. A prime number has exactly two factors, 1 and itself; the calculator tells you when your number is prime. And 1 is a special case with a single factor, so it is neither prime nor composite.
Common mistakes
- Forgetting 1 and the number itself. They are always factors, and a homework answer that leaves them out is marked wrong.
- Stopping too early. If you only test up to half the number instead of the square root you will still get the right answer, but if you stop at, say, 10 you will miss 12, 15 and 18 for 360.
- Listing a square root twice. For 36, the pair 6 × 6 counts 6 only once, giving 9 factors, not 10.
- Writing the prime factorization without exponents. 2 × 2 × 2 × 3 × 3 × 5 is correct, but teachers usually want 2³ × 3² × 5.
Where factors show up next
Factor lists are the foundation for the greatest common factor (the biggest factor two numbers share), the least common multiple (built from the highest prime powers), simplifying fractions, and factoring quadratics, where you look for a pair of factors of the constant term that add up to the middle coefficient. If that last one is what you are working on, the quadratic formula calculator will solve the equation and show the steps.
Frequently asked questions
›What is a factor of a number?
A factor is a whole number that divides another number with no remainder. 6 is a factor of 360 because 360 ÷ 6 = 60 exactly. Every number greater than 1 has at least two factors, 1 and itself.
›How do you find all the factors of a number?
Test every whole number from 1 up to the square root of the number. Each one that divides evenly gives you a pair: the divisor and the number divided by it. For 360, testing 1 through 18 produces 12 pairs and therefore 24 factors.
›What is the difference between factors and prime factorization?
The factor list is every number that divides evenly: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12 and so on for 360. The prime factorization is the one way to write the number as a product of primes: 360 = 2³ × 3² × 5.
›How many factors does a number have?
Add 1 to each exponent in the prime factorization and multiply. 360 = 2³ × 3² × 5¹ has (3 + 1)(2 + 1)(1 + 1) = 24 factors. A prime number always has exactly 2.
›Is 1 a prime number?
No. A prime needs exactly two factors, 1 and itself, and 1 has only one. It is neither prime nor composite.
›What is the largest number this factor calculator accepts?
1,000,000,000,000 (one trillion). Trial division only has to test up to the square root, one million, so results are still instant.