How to use the factor calculator
- Enter a whole number from 1 to 1 trillion, or tap one of the examples.
- Read the factor count and the list. The large figure is how many factors the number has, with a note saying whether it is prime or composite. Below it, every factor is listed in order.
- Check the pairs and the prime factorization. The factor pairs on the left multiply to your number. The cards show the prime factorization in exponent form, the sum of the factors, and whether the number is a perfect square.
How to find all the factors of a number
A factor of a number divides it with no remainder. OpenStax’s Prealgebra 2e (section 2.4) uses a simple systematic method: divide by 1, 2, 3 and so on, and every time the quotient is a whole number, the divisor and the quotient are a factor pair. Stop when the quotient becomes smaller than the divisor, because from then on the pairs repeat.
For 72, the textbook’s example:
| Divisor | Quotient | Factor pair |
|---|---|---|
| 1 | 72 | 1 × 72 |
| 2 | 36 | 2 × 36 |
| 3 | 24 | 3 × 24 |
| 4 | 18 | 4 × 18 |
| 6 | 12 | 6 × 12 |
| 8 | 9 | 8 × 9 |
5 and 7 don’t divide 72 evenly, and the next divisor that does, 9, would give a quotient of 8, which is already in the list. So 72 has 12 factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
The stopping point is the square root. √72 ≈ 8.49, so only divisors up to 8 need checking. That is why the calculator handles numbers up to a trillion quickly: it needs at most a million trial divisions.
Prime or composite?
A prime number has exactly two factors, 1 and itself. The primes below 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. A composite number has more. The number 1 has only one factor and is neither.
To test a number, the textbook divides by each prime in turn (2, 3, 5, 7, 11, …) and stops once the quotient is smaller than the divisor. For 83, none of 2, 3, 5 or 7 divides evenly, and 83 ÷ 11 is already less than 11, so 83 is prime. For 77, the test stops at 7: 77 = 7 × 11, so 77 is composite.
Quick divisibility checks help when you work by hand. A number is divisible by 2 if its last digit is even, by 3 if its digits add up to a multiple of 3, by 5 if it ends in 0 or 5, and by 10 if it ends in 0.
Prime factorization and counting factors
Every whole number greater than 1 can be written as a product of primes in exactly one way, apart from order. Section 2.5 of the same textbook works out 48 = 2⁴ × 3, 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5.
The prime factorization tells you how many factors there are without listing them. Add 1 to each exponent and multiply:
72 = 2³ × 3², so it has (3 + 1)(2 + 1) = 12 factors, matching the list. Each factor picks a power of 2 from 2⁰ to 2³ and a power of 3 from 3⁰ to 3²; there are 4 × 3 ways to choose.
A number has an odd number of factors only when it is a perfect square, because the square root pairs with itself. 36 has 9 factors, with 6 × 6 as the middle pair.
Sum of factors and perfect numbers
The sum of all factors of 72 is 195. Leaving out the number itself gives the sum of its proper factors, 123. When the proper factors add up to exactly the number, it is called a perfect number. 6 (1 + 2 + 3) and 28 (1 + 2 + 4 + 7 + 14) are the first two; the next are 496 and 8,128. The calculator flags them.
Where factors are useful
- Simplifying fractions: divide the top and bottom by a common factor. The simplify fractions calculator does this in one step.
- Greatest common factor and least common multiple: list or factor both numbers. The GCF and LCM calculators handle several numbers at once.
- Arranging things in equal groups: 72 chairs can be set out in rows of 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 or 72. The factor pairs give every possible rectangle.
- Area puzzles: a 36-square-foot rectangle with whole-number sides can be 1 × 36, 2 × 18, 3 × 12, 4 × 9 or 6 × 6.
Factors versus multiples
Factors and multiples are easy to mix up. The factors of 12 are the numbers that divide into it: 1, 2, 3, 4, 6 and 12, a short and finite list. The multiples of 12 are what you get by multiplying it: 12, 24, 36, 48 and so on, without end. If a is a factor of b, then b is a multiple of a.
Numbers with many factors are convenient for dividing things evenly. 60 has 12 factors, so an hour splits evenly into halves, thirds, quarters, fifths, sixths, tenths and twelfths. 360 has 24 factors, so a full circle divides into many whole-degree parts. By contrast 100 has only 9 factors, and a trillion, 2¹² × 5¹², has 169.
For a large number, the factor count comes from the exponent rule rather than a list: 1,000,000 = 2⁶ × 5⁶ has (6 + 1)(6 + 1) = 49 factors.
Common factors of two numbers
Listing the factors of two numbers side by side shows the factors they share. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. The factors of 72 are listed above. The numbers in both lists are 1, 2, 3, 4, 6, 8, 12 and 24, and the largest of them, 24, is the greatest common factor. Listing works well for small numbers; for larger ones, comparing prime factorizations is faster, and the GCF calculator does it for you. The smallest number that both 48 and 72 divide into, their least common multiple, is 144.
Sources & assumptions
How this calculator gets its numbers. Full method: our methodology.
- Factors are found by trial division up to the square root, pairing each divisor with its quotient.
- Whole numbers from 1 to 1 trillion; 1 has one factor and is neither prime nor composite.
- The prime factorization uses exponents (72 = 2³ × 3²); the number of factors is the product of each exponent plus one.
Sources
- OpenStax (Rice University), Prealgebra 2e, 2.4 Find Multiples and Factors. Finding all factors by dividing by counting numbers and pairing each divisor with its quotient (Example 2.46: the factors of 72), and testing for primes up to the point where the quotient is smaller than the divisor (Example 2.47: 83 is prime, 77 is composite).
- OpenStax (Rice University), Prealgebra 2e, 2.5 Prime Factorization and the Least Common Multiple. Prime factorizations of 48, 84 and 120 (Examples 2.48–2.50) and the LCM by listing multiples (15 and 20 → 60) and by prime factors (15 and 18 → 90; 50 and 100 → 100), Examples 2.52–2.54.
Last reviewed . For the greatest common factor or least common multiple of several numbers, use the GCF and LCM calculators.
Frequently asked questions
What are the factors of 72?
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72: twelve factors in six pairs, from 1 × 72 to 8 × 9.
How do I find all the factors of a number?
Divide by 1, 2, 3 and so on up to the square root. Each divisor that leaves no remainder gives a factor pair with its quotient.
Is 1 a prime number?
No. A prime has exactly two factors, 1 and itself. The number 1 has only one factor, so it is neither prime nor composite.
What is the difference between factors and prime factors?
Factors are all the whole numbers that divide a number evenly. Prime factors are only the primes among them, written with exponents: 72 has 12 factors but its prime factorization is 2³ × 3².
How can I tell how many factors a number has?
Write the prime factorization, add 1 to each exponent and multiply. 72 = 2³ × 3² has (3 + 1)(2 + 1) = 12 factors.
Why does a perfect square have an odd number of factors?
Factors come in pairs, except the square root, which pairs with itself. 36 has 9 factors because 6 × 6 counts once.