Factor Calculator
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Results
360 has 24 factors
All 24 factors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 How they were found 1. Prime factorization: 360 = 2³ × 3² × 5 2. Number of factors: add 1 to each exponent and multiply: (3 + 1) × (2 + 1) × (1 + 1) = 24 3. Sum of factors: (2⁴ − 1) ÷ (2 − 1) × (3³ − 1) ÷ (3 − 1) × (1 + 5) = 15 × 13 × 6 = 1170 4. Factor pairs: every factor d up to √360 ≈ 18.97 pairs with 360 ÷ d, as in the table. 5. Sum of proper factors (all except 360) = 1170 − 360 = 810, which is > 360, so 360 is abundant.
| Pair | Factor | Paired factor |
|---|---|---|
| 1 | 1 | 360 |
| 2 | 2 | 180 |
| 3 | 3 | 120 |
| 4 | 4 | 90 |
| 5 | 5 | 72 |
| 6 | 6 | 60 |
| 7 | 8 | 45 |
| 8 | 9 | 40 |
| 9 | 10 | 36 |
| 10 | 12 | 30 |
| 11 | 15 | 24 |
| 12 | 18 | 20 |
How to use the factor calculator
- Enter a whole number from 1 up to 18 digits (below 1018).
- Read the number of factors, the prime factorization and the classification in the results.
- Copy the full list of factors from the working, and scroll the factor-pairs table to see each pair that multiplies to your number.
Factor formulas
Number of factors d(n) = (a + 1)(b + 1)(c + 1) …
Sum of factors σ(n) = (p₁a+1 − 1)/(p₁ − 1) × (p₂b+1 − 1)/(p₂ − 1) × …
A factor (or divisor) of n is a whole number that divides n with no remainder. Here p₁, p₂, p₃ are the different primes in n and a, b, c are their exponents. Every factor of n is made by choosing a power of each prime, from the zeroth power up to its exponent, which is why the number of factors is a product of (exponent + 1) terms. The calculator first finds the prime factorization exactly, using trial division by small primes, the Miller–Rabin primality test and Pollard's rho method for large factors, and then builds every factor from it. This is far quicker than testing every number up to n.
Worked example: the factors of 360
360 = 2³ × 3² × 5, so it has (3 + 1) × (2 + 1) × (1 + 1) = 24 factors. You only need to test divisors up to √360 ≈ 18.97, because each one pairs with a partner above it. The 12 factor pairs are 1 × 360, 2 × 180, 3 × 120, 4 × 90, 5 × 72, 6 × 60, 8 × 45, 9 × 40, 10 × 36, 12 × 30, 15 × 24, 18 × 20.
The factors add up to 1,170. The proper factors (every factor except 360 itself) add up to 810, which is more than 360, so 360 is an abundant number.
Factors of common numbers
| Number | Factors | Count | Sum | Type |
|---|---|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 6 | 28 | abundant |
| 16 | 1, 2, 4, 8, 16 | 5 | 31 | deficient |
| 18 | 1, 2, 3, 6, 9, 18 | 6 | 39 | abundant |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 | 60 | abundant |
| 30 | 1, 2, 3, 5, 6, 10, 15, 30 | 8 | 72 | abundant |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 9 | 91 | abundant |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 10 | 124 | abundant |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 12 | 168 | abundant |
| 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 | 12 | 195 | abundant |
| 84 | 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 | 12 | 224 | abundant |
| 90 | 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 | 12 | 234 | abundant |
| 96 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 | 12 | 252 | abundant |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 9 | 217 | abundant |
| 120 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 | 16 | 360 | abundant |
Numbers with record numbers of factors
Each of these has more factors than any smaller number (they are called highly composite numbers):
| Number | Factors |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 4 | 3 |
| 6 | 4 |
| 12 | 6 |
| 24 | 8 |
| 36 | 9 |
| 48 | 10 |
| 60 | 12 |
| 120 | 16 |
| 180 | 18 |
| 240 | 20 |
| 360 | 24 |
| 720 | 30 |
| 840 | 32 |
| 1,260 | 36 |
| 1,680 | 40 |
| 2,520 | 48 |
| 5,040 | 60 |
Reading the results
- Prime or composite. A prime has exactly two factors, 1 and itself. A composite has more. The number 1 has a single factor and is neither.
- Perfect, abundant or deficient. Compare the sum of the proper factors with the number. The perfect numbers below 10,000 are 6, 28, 496, 8128. Every prime is deficient, because its only proper factor is 1.
- Perfect squares have an odd number of factors. Their square root pairs with itself: 36 = 6 × 6, so 36 has 9 factors rather than an even count.
Tips
- Divisibility shortcuts. A number is divisible by 3 if its digits add up to a multiple of 3, by 4 if its last two digits form a multiple of 4, and by 9 if its digit sum is a multiple of 9.
- Factors come in pairs. When listing by hand, write each small factor together with its partner and stop at the square root, so you never miss one.
- For the prime factors alone, drawn as a factor tree or division ladder, use the prime factorization calculator. For the factors two or more numbers share, use the common factor calculator or the GCF calculator.
Frequently asked questions
What are the factors of 360?
360 has 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 and 360. They form 12 factor pairs, from 1 × 360 to 18 × 20, and add up to 1,170.
How do I count the factors of a number without listing them?
Write the prime factorization, add 1 to each exponent and multiply. 360 = 2³ × 3² × 5, so it has (3 + 1) × (2 + 1) × (1 + 1) = 24 factors. Each factor picks a power of 2 (four choices), a power of 3 (three) and a power of 5 (two).
What is the difference between a factor and a multiple?
A factor divides a number exactly; a multiple is the number times a whole number. 6 is a factor of 18, and 18 is a multiple of 6. A number has a limited list of factors but infinitely many multiples.
What is a perfect number?
A number whose proper factors (all factors except itself) add up to the number. 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28. The first four perfect numbers are 6, 28, 496 and 8,128. If the proper factors add up to more, the number is abundant (12: 1 + 2 + 3 + 4 + 6 = 16); if less, it is deficient.
How big a number can the factor calculator handle?
Any whole number up to 18 digits (below 10¹⁸). It is factored exactly with trial division, the Miller–Rabin prime test and Pollard's rho method, then every factor is built from the primes. Lists are written out for up to 10,000 factors; the number below 10¹² with the most factors, 963,761,198,400, has 6,720.