The factors of a number are the positive whole numbers that divide it with no remainder. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36, nine in all. The list comes out sorted with the total underneath, and it helps you decide whether a number is prime, split things into equal groups, and compare two numbers before looking for their greatest common factor. If you want to write the number as a product of primes, use prime factorization; here you get every factor.
Worked example: the factors of 36
You want to set up 36 chairs in full rows, each with the same number of chairs. How many row arrangements are possible?
- Test 1 through 6. Since √36 = 6, there is no need to go past it. The numbers 1, 2, 3, 4, and 6 divide 36.
- Find each one’s partner. 36 ÷ 1 = 36, 36 ÷ 2 = 18, 36 ÷ 3 = 12, 36 ÷ 4 = 9, and 36 ÷ 6 = 6.
- Combine and drop the repeat. The 6 shows up twice because 6 × 6 = 36, and it is counted only once.
- Sort. 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Result: 9 factors, which means 9 ways to make the rows.
Check: 36 = 2² × 3², so (2 + 1)(2 + 1) = 9, the same total.
Counting factors from the exponents
To find how many factors a number has, you do not need to list them. Factor the number, add 1 to each exponent, and multiply the results. For 360 = 2³ × 3² × 5 you get (3 + 1)(2 + 1)(1 + 1) = 24. This works because each factor picks, for every prime, an exponent between 0 and the one in the number, and the choices multiply. The calculator builds the list with this method, checks that every value divides the number, and confirms that the total matches the count. If you need the exponents, the prime factorization calculator gives them.
Primes, squares, and perfect numbers
| Type | Example | What you see in the list |
|---|---|---|
| Prime | 13 | Only 1 and 13 |
| Perfect square | 100 | 9 factors, an odd count because of the pair 10 × 10 |
| Power of 2 | 64 | 7 factors, all powers of 2 |
| Perfect number | 28 | 1 + 2 + 4 + 7 + 14 = 28, the sum of the other factors |
Perfect numbers are rare, and after 28 comes 496. The case that confuses people most is the square, because its middle pair does not contribute two different factors.
Common factors and the GCF
The common factors of two numbers are the ones that appear in both lists, and the largest of them is the GCF. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Comparing them with those of 36, the shared ones are 1, 2, 3, 4, 6, and 12, so the GCF of 36 and 60 is 12. You can use it to simplify: 36/60 = 3/5, dividing the top and bottom by 12. This calculator lists one number at a time; for the GCF of several numbers directly, use the greatest common factor calculator, and for common multiples, the least common multiple calculator, because a number’s factors run out and its multiples do not. To reduce a fraction in one step, there is simplify fractions.
Before listing, the divisibility rules tell you at a glance whether 2, 3, 5, 9, or 10 are factors.
Factors of common numbers
| Number | Factors | Total |
|---|---|---|
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 |
| 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 | 12 |
| 120 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 | 16 |
| 180 | 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 | 18 |
The more small factors a number has, the more ways it can be split. The numbers 24 and 180 can be split into 8 and 18 equal parts, respectively.
Zero and negative numbers
Zero is not accepted because every nonzero integer divides it, so its list would never end. A negative such as −36 has the same factors as 36 with a minus sign, so enter 36 and add the sign yourself. If you enter 0 or a negative number, the calculator returns no list and shows an invalid-input notice.
Frequently asked questions
Does the list include 1 and the number itself?
Yes. 1 and the number itself always divide n, so every list starts at 1 and ends at n. A number with only those two factors is prime: 23 has 1 and 23. The number 1 is the exception, because its list has a single entry, and that is why it is not considered prime.
Why does a perfect square have an odd number of factors?
Factors pair up so that each pair multiplies to n. In a square, one pair is the root with itself and is not repeated. For 49, the pairs are 1 × 49 and 7 × 7, and the factors are 1, 7, and 49: three, an odd number.
How do I count the factors without writing them all out?
Break the number into primes, add 1 to each exponent, and multiply. With 2310 = 2 × 3 × 5 × 7 × 11, every exponent is 1, so you get 2 × 2 × 2 × 2 × 2 = 32 factors. With a single repeated prime, such as 3⁴, it would be 4 + 1 = 5 factors.