Factor Calculator: All Factors of a Number
Type a whole number to list all of its factors and factor pairs, with the count worked out from its prime factorization.
Any whole number up to 20 digits.
Factors of 120
16factors
120 has 16 factors.
- Factor pairs
- 8
- Prime factors
- 2³ × 3 × 5
- Sum of factors
- 360
How to find all the factors of a number
A factor of n divides it exactly, and factors come in pairs that multiply to n: 120 = 1 × 120 = 2 × 60 = 3 × 40 = 4 × 30 = 5 × 24 = 6 × 20 = 8 × 15 = 10 × 12. To list them by hand, try 1, 2, 3 … up to the square root of n (about 10.95 for 120); every factor you find there brings its partner from above.
The prime factorization tells you how many factors there will be before you list them. 120 = 2³ × 3 × 5, and a factor can use 2 zero to three times (4 choices), 3 zero or one time (2 choices) and 5 zero or one time (2 choices), so 120 has 4 × 2 × 2 = 16 factors.
Only perfect squares have an odd number of factors, because the square root pairs with itself: 36 has 9 factors (1, 2, 3, 4, 6, 9, 12, 18, 36), with 6 × 6 as its middle pair. A prime has exactly two factors.
n = p₁^a₁ × p₂^a₂ × … ⇒ number of factors = (a₁ + 1)(a₂ + 1)…
- Grade 4 students find all factor pairs of whole numbers from 1 to 100 and use them to tell primes from composites (4.OA.B.4). Source: Common Core State Standards, Grade 4 Operations and Algebraic Thinking (4.OA.B.4).
- Numbers with more factors than any smaller number are called highly composite: 12, 24, 36, 48, 60, 120, 180, 240, 360, 720 … Source: OEIS A002182, highly composite numbers.
- A perfect number equals the sum of its other factors: 6 = 1 + 2 + 3, and the next are 28, 496 and 8,128. Source: OEIS A000396, perfect numbers.
Numbers with the most factors
Highly composite numbers: each has more factors than any smaller number, which is why 12, 60 and 360 are popular for clocks, minutes and degrees.
| Number | Factors | Prime factorization |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 2 |
| 4 | 3 | 2² |
| 6 | 4 | 2 × 3 |
| 12 | 6 | 2² × 3 |
| 24 | 8 | 2³ × 3 |
| 36 | 9 | 2² × 3² |
| 48 | 10 | 2⁴ × 3 |
| 60 | 12 | 2² × 3 × 5 |
| 120 | 16 | 2³ × 3 × 5 |
Show all 21 rowsShow fewer
| 180 | 18 | 2² × 3² × 5 |
| 240 | 20 | 2⁴ × 3 × 5 |
| 360 | 24 | 2³ × 3² × 5 |
| 720 | 30 | 2⁴ × 3² × 5 |
| 840 | 32 | 2³ × 3 × 5 × 7 |
| 1,260 | 36 | 2² × 3² × 5 × 7 |
| 1,680 | 40 | 2⁴ × 3 × 5 × 7 |
| 2,520 | 48 | 2³ × 3² × 5 × 7 |
| 5,040 | 60 | 2⁴ × 3² × 5 × 7 |
| 7,560 | 64 | 2³ × 3³ × 5 × 7 |
| 10,080 | 72 | 2⁵ × 3² × 5 × 7 |
Frequently Asked Questions
What are the factors of 24?
1, 2, 3, 4, 6, 8, 12 and 24: eight factors in four pairs (1 × 24, 2 × 12, 3 × 8, 4 × 6). From 24 = 2³ × 3, the count is (3 + 1)(1 + 1) = 8.
What is the difference between factors and multiples?
Factors divide into a number; multiples are what you get by multiplying it. 6 is a factor of 24, and 24 is a multiple of 6. A number has finitely many factors but infinitely many multiples.
Is a number a factor of itself?
Yes. Every whole number n has 1 and n as factors. The proper factors leave n itself out; adding those is how perfect, abundant and deficient numbers are defined.