Factor Calculator
This factor calculator lists every factor of a whole number in order, from 1 up to the number itself. It also lays out the factor pairs, counts the factors, adds them up, says whether the number is prime, composite or neither, flags perfect squares and gives the prime factorization. Type any whole number from 1 to one trillion and the steps show which divisors were tested.
How to use the factor calculator
- Type a whole number into the box. The calculator starts with 36 so you can see a full result straight away.
- Read the factors at the top, listed from smallest to largest. For 36 that is
1, 2, 3, 4, 6, 9, 12, 18, 36. - Check the factor pairs table underneath. Each row is two numbers that multiply to give your number: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6.
- Look at the summary lines: 36 has 9 factors, they add up to 91, it is composite, it is a perfect square, and its prime factorization is
2² × 3². - Open the steps to see each divisor tested from 1 up to the square root.
The box accepts whole numbers from 1 to 1,000,000,000,000. If you want the prime building blocks drawn as a tree or a division ladder, the prime factorization calculator shows that working in full.
Factors are times-table facts read backwards: every spot where a number appears in a multiplication chart is one of its factor pairs.
How to find the factors of a number
A factor of a number divides it exactly, leaving a remainder of 0. So 4 is a factor of 36 because 36 ÷ 4 = 9 with nothing left over, and 5 is not because 36 ÷ 5 = 7 remainder 1. Factors are also called divisors; the two words mean the same thing here.
That partner rule is what makes the method fast. You do not need to test every number up to n. You only test up to the square root of n, because every factor above the square root is the partner of one below it.
- Work out
√n, or at least the whole number just below it. For 36,√36 = 6. - Try each whole number from 1 up to that limit. Each one that divides evenly gives you a pair.
- Write the small factors in a row going up and their partners in a row going down, then read them off in order.
Why stop at the square root? If two numbers multiply to 36 and both were larger than 6, their product would be larger than 6 × 6 = 36. So every pair has a member of 6 or less, and testing up to 6 catches them all. Near one trillion, that means about a million tests instead of a trillion.
By hand, divisibility checks save time: 2 if the last digit is even, 3 if the digits add to a multiple of 3, 5 if it ends in 0 or 5, 9 if the digits add to a multiple of 9. For anything else, a remainder calculator tells you instantly whether the division comes out even.
Worked examples
Example: factors of 48
The square root of 48 is about 6.93, so test 1 through 6.
48 has 10 factors in 5 pairs. Its prime factorization is 2⁴ × 3.
Example: factors of 97
The square root of 97 is about 9.85, so test 1 through 9.
Only 1 and 97 divide 97, so 97 is prime. A prime always has exactly two factors. For more on how to test a number this way, see what a prime number is.
Example: factors of 100
√100 = 10 exactly, so test 1 through 10.
100 has 9 factors, an odd count, because 10 pairs with itself and is only listed once. The factors add up to 217.
Factors of common numbers
The factors of 12, 24, 36, 48, 60 and 72 come up constantly: in times tables, in simplifying fractions, and in any question about splitting things into equal groups. Here they are with the number of factors, the number of factor pairs and the prime factorization.
| Number | Factors | How many | Pairs | Prime factorization |
|---|---|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 6 | 3 | 2² × 3 |
| 16 | 1, 2, 4, 8, 16 | 5 | 3 | 2⁴ |
| 18 | 1, 2, 3, 6, 9, 18 | 6 | 3 | 2 × 3² |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 | 4 | 2³ × 3 |
| 30 | 1, 2, 3, 5, 6, 10, 15, 30 | 8 | 4 | 2 × 3 × 5 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 9 | 5 | 2² × 3² |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 10 | 5 | 2⁴ × 3 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 12 | 6 | 2² × 3 × 5 |
| 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 | 12 | 6 | 2³ × 3² |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 9 | 5 | 2² × 5² |
| 120 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 | 16 | 8 | 2³ × 3 × 5 |
| 144 | 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 | 15 | 8 | 2⁴ × 3² |
Notice which rows have an odd number of factors: 16, 36, 100 and 144. Those are the perfect squares in the table. The pattern holds for every whole number, and the next section explains why.
Counting factors from the prime factorization
You can count the factors of a number without listing them. Write the number as a product of prime powers, add 1 to each exponent, and multiply the results.
It works because every factor of 72 is built by choosing how many 2s to use (0, 1, 2 or 3 of them, four choices) and how many 3s (0, 1 or 2, three choices). Each combination gives a different factor, and there are 4 × 3 = 12 combinations.
The formula also explains the perfect-square pattern. The count is odd only when every (exponent + 1) is odd, so every exponent is even, and that is exactly what a perfect square is. In pair terms, only a perfect square has a pair like 6 × 6 whose matching halves count once.
The same perfect-square test is what the square root calculator relies on when it pulls the largest square factor out of a root, such as √72 = 6√2. Factor lists are also how you reduce a fraction: divide the top and bottom by a shared factor, or let the simplify fractions calculator find the greatest one.
Prime, composite or neither
The factor count sorts every whole number into one of three groups, and the calculator reports which one yours falls into.
- Prime: exactly two factors, 1 and itself. Examples: 2, 13, 97.
- Composite: three or more factors. Examples: 4, 36, 100.
- Neither: the number 1, which has only one factor.
The reasons 1 sits in its own group, and the quickest ways to tell primes and composites apart, are covered in prime and composite numbers.
The sum of factors is reported because of a much older idea. Leave out the number itself and add the rest. If the total equals the number, it is called a perfect number. 6 qualifies (1 + 2 + 3 = 6) and so does 28 (1 + 2 + 4 + 7 + 14 = 28). For 36 the factors other than 36 add to 55, which is more than 36, so it is not perfect.
From factors to common factors
When you list the factors of two numbers side by side, the numbers in both lists are their common factors, and the largest of them is the greatest common factor. For 36 and 48 the common factors are 1, 2, 3, 4, 6 and 12, so the GCF is 12. The GCF calculator handles that for up to 20 numbers at once, and the LCM calculator works in the other direction, finding the smallest shared multiple. Both methods are compared in how to find the GCF and LCM.
Common mistakes
- Forgetting 1 and the number itself. Every whole number above 1 has at least these two factors. Leaving them out is the most frequent slip in factor lists.
- Stopping too early. Testing only up to half of n wastes effort, but stopping before the square root misses pairs. For 48, stopping at 4 loses the pair 6 × 8.
- Listing the square root twice. In 36, the pair 6 × 6 contributes one factor, not two. That is why perfect squares have an odd count.
- Mixing up factors and prime factors. 36 has nine factors but only two prime factors, 2 and 3. A question asking for “the factors of 36” wants all nine.
- Confusing factors with multiples. 72 is a multiple of 36, not a factor of it. Factors are never larger than the number.
Frequently asked questions
What are the factors of 24?
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. That is eight factors, arranged in four pairs: 1 × 24, 2 × 12, 3 × 8 and 4 × 6. The square root of 24 is about 4.9, so testing 1 to 4 finds every pair. The prime factorization is 2³ × 3, and (3 + 1)(1 + 1) = 8 confirms the count.
What are the factors of 72?
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72. That makes twelve factors in six pairs: 1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12 and 8 × 9. They add up to 195. Because 72 = 2³ × 3², the count follows from (3 + 1)(2 + 1) = 12.
What is a factor pair?
A factor pair is two whole numbers that multiply to give a particular number. The factor pairs of 18 are 1 × 18, 2 × 9 and 3 × 6. Every factor belongs to exactly one pair, so finding the small half of each pair, up to the square root, gives you the whole list. A perfect square has one pair made of the same number twice, such as 5 × 5 for 25.
How many factors does 100 have?
100 has nine factors: 1, 2, 4, 5, 10, 20, 25, 50 and 100. The count is odd because 100 is a perfect square, and its middle factor 10 pairs with itself. You can confirm the count from the prime factorization 100 = 2² × 5²: add one to each exponent and multiply, 3 × 3 = 9.
Is 1 a factor of every number?
Yes. 1 divides every whole number exactly, so it is always the first factor in the list, and the number itself is always the last. That is why the smallest possible factor list for a number above 1 has two entries, which is the definition of a prime. The number 1 is the only whole number with just one factor.
What is the difference between factors and multiples?
Factors divide into a number and multiples are produced by multiplying it. The factors of 6 are 1, 2, 3 and 6, a finite list that never goes above 6. The multiples of 6 are 6, 12, 18, 24 and onward without end. If 3 is a factor of 6, then 6 is a multiple of 3; the two words describe one relationship from opposite sides.
How do you know if a number is prime from its factors?
A number is prime when its factor list contains exactly two entries, 1 and the number itself. If the calculator finds any other factor between 2 and the square root, the number is composite. For example, 91 looks prime but 91 = 7 × 13, so its factors are 1, 7, 13 and 91 and it is composite. The number 1 is neither.
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