How to use the factor calculator
- Enter a whole number from 1 to 1,000,000,000,000. Decimals, negative numbers and 0 get a short message instead of an answer.
- The top of the results shows how many factors the number has, and whether it is prime or composite.
- Copy the full list from the All factors box, or read the factor pairs table, where each row multiplies to your number.
- The working shows how the factors were found and how the count follows from the prime factorization.
How factors are found
A factor of n is a whole number that divides n with no remainder. Factors come in pairs: if a divides n, so does n ÷ a. One number of each pair is always at most √n, so you only need to test up to the square root.
The second formula comes from the prime factorization. Each factor picks how many copies of each prime to take, from 0 up to its exponent, so a prime with exponent e offers e + 1 choices. Multiplying the choices gives the count without listing anything.
Worked example
The calculator starts with 120.
- √120 ≈ 10.95, so test 1 to 10.
- The ones that divide 120 evenly are 1, 2, 3, 4, 5, 6, 8 and 10. Seven and nine leave remainders.
- Each gives a pair: 1 × 120, 2 × 60, 3 × 40, 4 × 30, 5 × 24, 6 × 20, 8 × 15 and 10 × 12.
- That's 8 pairs, so 16 factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120.
- Check with the prime factorization 120 = 2³ × 3 × 5: (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16.
The factors add up to 360. Leave out 120 itself and the proper factors add up to 240, which is more than 120, so 120 is an abundant number.
Prime, perfect, abundant and deficient
| Type | Rule | Examples |
|---|---|---|
| Prime | Exactly two factors, 1 and itself | 2, 3, 5, 7, 97 |
| Perfect square | An odd number of factors, because one pair repeats | 36 (6 × 6), 100 |
| Perfect | Proper factors add up to the number | 6, 28, 496, 8,128 |
| Abundant | Proper factors add up to more | 12, 18, 20, 120 |
| Deficient | Proper factors add up to less | 8, 10, 97, all primes |
The number 1 has one factor and is neither prime nor composite. For 28, the proper factors are 1, 2, 4, 7 and 14, and 1 + 2 + 4 + 7 + 14 = 28, which makes it perfect. Perfect numbers are rare: only a few dozen are known, and every one found so far is even.
Frequently asked questions
What is the difference between factors and prime factors?
Factors are all the numbers that divide n evenly, such as the 16 factors of 120. Prime factors are only the primes among them, multiplied together to rebuild the number: 120 = 2 × 2 × 2 × 3 × 5. The prime factorization calculator shows that breakdown step by step.
Why does a perfect square have an odd number of factors?
Factors normally come in pairs of two different numbers. For a perfect square, one pair is the square root times itself, such as 6 × 6 = 36, so that factor is counted once. Every other factor still has a partner, which leaves an odd total.
How many factors can a number have?
It depends on the exponents in its prime factorization, not on how big the number is. 1,000,000,000,000 = 2¹² × 5¹² has 13 × 13 = 169 factors, while 720,720, a much smaller number, has 240. A prime of any size has just two.
Do negative numbers have factors?
Yes. Each positive factor also works with a minus sign, so −12 is divisible by ±1, ±2, ±3, ±4, ±6 and ±12. The calculator lists the positive factors; add the negatives yourself if you need them.
Related calculators
- Prime Factorization Calculator — Break a number into prime factors with exponents, see the factor tree, check if it is prime and count its divisors.
- Common Factor Calculator — Every factor shared by two or more whole numbers, with the greatest common factor and each number's factor list.
- GCF Calculator — Greatest common factor of two or more whole numbers, with Euclid's algorithm and a prime factorization check.
- LCM Calculator — Least common multiple of two or more whole numbers, worked out by prime factorization and by the GCF method.
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