Factor Calculator
List every factor of a whole number, with its factor pairs, prime factorization, factor count and sum, and each step shown.
Enter one whole number, such as 36 or 360. Commas are ignored.
36 is a perfect square, so its middle factor pair is 6 × 6 and the count is odd.
All factors of 36
Listed from smallest to largest. Each one divides the number with no remainder.
Proper factors (all except 36) add up to 55. That is more than 36, so 36 is an abundant number.
Your calculation
The same steps the calculator runs, with your number.
Factor pairs
Each row multiplies to the number. A highlighted row is a square root pair.
| Pair | Factor | Partner |
|---|
Prime factorization by division
Divide by the smallest prime that fits until the quotient is 1.
| Step | Division | Quotient |
|---|
Whole numbers up to 10¹² are factored exactly by trial division. Numbers with more than 120 factors show the smallest and largest 60.
What Is a Factor?
A factor of a whole number divides it with no remainder. Factors are also called divisors. Because 72 = 8 × 9, both 8 and 9 are factors of 72, and 72 is a multiple of each of them.
Factors come in factor pairs that multiply to the number. The pairs for 24 are 1 × 24, 2 × 12, 3 × 8 and 4 × 6, so 24 has 8 factors: 1, 2, 3, 4, 6, 8, 12 and 24. Every number is a factor of itself, and 1 is a factor of every number.
A prime number is a counting number greater than 1 whose only factors are 1 and itself, such as 7 or 97. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
How to Use the Factor Calculator
- Enter a number. Type any whole number from 1 to 1,000,000,000,000. Commas such as 1,000 are fine. A negative number uses the factors of its absolute value.
- Read the count. The large number is how many positive factors the number has. The chips say whether it is prime or composite, and flag perfect squares and perfect numbers.
- Use the details. The next card lists every factor in order and says whether the number is perfect, abundant or deficient. Below it are the prime factorization, the number of factor pairs, the sum of factors, the square root, the steps with your number, a factor pair table and the division steps.
Results update as you type, and Enter recalculates. The quick examples include common homework numbers and a prime.
How to Find All Factors of a Number
Divide the number by each whole number, starting at 1. When the quotient is a whole number, the divisor and the quotient are a factor pair. Stop when the divisor passes the square root, because after that point the pairs repeat in reverse order.
Worked example: factors of 96
√96 ≈ 9.798, so test the divisors 1 to 9. The ones that divide 96 evenly are 1, 2, 3, 4, 6 and 8, giving the pairs 1 × 96, 2 × 48, 3 × 32, 4 × 24, 6 × 16 and 8 × 12. The divisors 5, 7 and 9 leave a remainder.
Write the small members in order, then the large members in reverse: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96. That is all 12 factors of 96, found with only 9 test divisions.
Divisibility Tests
These tests show whether a number divides evenly without doing the division.
| Divisible by | Test |
|---|---|
| 2 | Last digit is 0, 2, 4, 6 or 8 |
| 3 | Sum of the digits is divisible by 3 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 10 | Last digit is 0 |
Example: 2,346 ends in 6, so it is divisible by 2. Its digits add to 2 + 3 + 4 + 6 = 15, so it is divisible by 3 and therefore by 6: 2,346 ÷ 6 = 391. It does not end in 0 or 5, so 5 and 10 are not factors.
Prime Factorization
Every whole number above 1 is a product of primes in exactly one way, apart from the order. This is the fundamental theorem of arithmetic. To find it, divide by the smallest prime that fits and repeat on the quotient.
The prime factorization drives both formulas below and is the starting point for the GCF and LCM.
Number of Factors Formula
Each factor picks one power of every prime, from power 0 up to its exponent. The prime 2 offers 4 choices (2⁰ to 2³), 3 offers 3 and 5 offers 2, so there are 24 combinations.
Sum of Factors Formula
σ(n) adds every factor, including n itself. Subtract n to get the sum of the proper factors, which decides whether a number is perfect, abundant or deficient.
Factors of Common Numbers
Factor lists for numbers that often come up in homework, fractions and measurements.
| Number | Factors | Count | Prime factorization |
|---|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 6 | 2² × 3 |
| 16 | 1, 2, 4, 8, 16 | 5 | 2⁴ |
| 18 | 1, 2, 3, 6, 9, 18 | 6 | 2 × 3² |
| 20 | 1, 2, 4, 5, 10, 20 | 6 | 2² × 5 |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 | 2³ × 3 |
| 30 | 1, 2, 3, 5, 6, 10, 15, 30 | 8 | 2 × 3 × 5 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 9 | 2² × 3² |
| 40 | 1, 2, 4, 5, 8, 10, 20, 40 | 8 | 2³ × 5 |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 10 | 2⁴ × 3 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 12 | 2² × 3 × 5 |
| 64 | 1, 2, 4, 8, 16, 32, 64 | 7 | 2⁶ |
| 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 | 12 | 2³ × 3² |
| 84 | 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 | 12 | 2² × 3 × 7 |
| 96 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 | 12 | 2⁵ × 3 |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 9 | 2² × 5² |
| 120 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 | 16 | 2³ × 3 × 5 |
| 144 | 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 | 15 | 2⁴ × 3² |
The rows with an odd count (16, 36, 64, 100 and 144) are exactly the perfect squares.
Perfect, Abundant and Deficient Numbers
Compare a number with the sum of its proper factors (every factor except the number itself):
- Perfect: the sum equals the number. 28 has proper factors 1, 2, 4, 7 and 14, which add to 28. The smallest perfect numbers are 6, 28, 496 and 8,128.
- Abundant: the sum is larger. For 12, 1 + 2 + 3 + 4 + 6 = 16. The smallest odd abundant number is 945.
- Deficient: the sum is smaller. For 8, 1 + 2 + 4 = 7. Every prime is deficient, because its only proper factor is 1.
Understanding Your Result
A count of exactly 2 means the number is prime. A count of 1 happens only for the number 1.
An odd count means the number is a perfect square. Every other number has an even count, because its factors split cleanly into pairs.
The square root tile shows where the pair search stops. No factor pair has both members above the square root.
Where Factors Are Used
Arranging items in equal rows
The factor pairs of 24 are every way to set out 24 chairs in a full rectangle: 1 row of 24, 2 rows of 12, 3 rows of 8 or 4 rows of 6, plus the same shapes turned sideways.
Splitting into equal groups
A class of 30 students divides evenly into groups of 1, 2, 3, 5, 6, 10, 15 or 30. Groups of 4 leave 2 students over, because 4 is not a factor of 30.
Simplifying fractions and ratios
The largest factor two numbers share is their greatest common factor. The factors of 24 and 36 have 12 as the largest shared value, so 24/36 simplifies to 2/3.
Dividing a circle or an hour
360 has 24 factors and 60 has 12, so a full turn of 360 degrees and an hour of 60 minutes split into many equal whole number parts, such as 15 degree or 15 minute steps.
Common Mistakes
- Forgetting the partner. When 4 divides 96, the quotient 24 is also a factor. Write down both numbers of every pair.
- Counting the square root twice. 6 × 6 = 36 adds only one factor, 6.
- Leaving out 1 or the number itself. Both are always factors.
- Mixing up factors and multiples. Factors of 12 are at most 12 (1, 2, 3, 4, 6, 12). Multiples of 12 are at least 12 (12, 24, 36, …).
- Treating every factor as a prime factor. 6 is a factor of 36 but not a prime factor. The prime factors of 36 are only 2 and 3.
- Calling 1 a prime. A prime needs exactly two factors, and 1 has only one.
Related Calculations
The GCF Calculator finds the largest factor that two or more numbers share, and the LCM Calculator finds their smallest common multiple. Both use the same prime factorization shown here.
The Fraction Calculator reduces fractions with common factors, the Root Calculator uses prime factors to simplify radicals such as √72 = 6√2, and the Exponent Calculator works with the prime powers in a factorization.
Frequently Asked Questions
What are the factors of 36?
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36, which is 9 factors. They come in the pairs 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. Because 36 is a perfect square, the pair 6 × 6 uses the same number twice, so 36 has an odd number of factors. The prime factorization is 36 = 2² × 3², and (2 + 1) × (2 + 1) = 9 confirms the count. The factors add up to 91, and the proper factors (all except 36) add up to 55.
How do you find all the factors of a number?
Divide the number by each whole number starting at 1 and stop at its square root. Every divisor that leaves no remainder gives a factor pair: the divisor and the quotient. For 84, √84 ≈ 9.165, so test 1 to 9. The divisors 1, 2, 3, 4, 6 and 7 give the pairs 1 × 84, 2 × 42, 3 × 28, 4 × 21, 6 × 14 and 7 × 12. Listing both members of each pair in order gives all 12 factors: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. Divisibility tests, such as the digit sum test for 3, speed up the checks.
What is the difference between factors and prime factors?
Factors are all the whole numbers that divide a number evenly, while prime factors are only the factors that are prime numbers. The factors of 12 are 1, 2, 3, 4, 6 and 12, but its prime factors are just 2 and 3. The prime factorization writes the number as a product of primes, 12 = 2² × 3, and it is unique for every whole number above 1. Every factor of 12 can be built from that factorization by choosing a power of 2 from 2⁰ to 2² and a power of 3 from 3⁰ to 3¹, which gives 3 × 2 = 6 factors.
How many factors does a number have?
Write the number as a product of prime powers, add 1 to each exponent, and multiply the results. For 360 = 2³ × 3² × 5, the count is (3 + 1) × (2 + 1) × (1 + 1) = 4 × 3 × 2 = 24 factors. A prime number always has exactly 2 factors, 1 and itself, and the number 1 has only 1 factor. Perfect squares such as 36 and 100 are the only numbers with an odd count, because their square root pairs with itself. Among numbers up to 100, the most factors any number has is 12, reached by 60, 72, 84, 90 and 96.
What is a factor pair?
A factor pair is two whole numbers that multiply to give the number. For 24 the factor pairs are 1 × 24, 2 × 12, 3 × 8 and 4 × 6. The smaller number of each pair is never larger than the square root, so you only need to test up to √24 ≈ 4.899. A number with an even count of factors has exactly half that many pairs, so 24 with 8 factors has 4 pairs. A perfect square has one pair made of the same number twice, such as 6 × 6 for 36, so 36 has 9 factors and 5 pairs.
Sources & References
- OpenStax. Prealgebra 2e, 2.4 Find Multiples and Factors (finding all factors, divisibility tests, prime and composite numbers). openstax.org
- OpenStax. Prealgebra 2e, 2.5 Prime Factorization and the Least Common Multiple. openstax.org
- NIST Digital Library of Mathematical Functions. §27.2 Functions (fundamental theorem of arithmetic, d(n), σ(n), table of values). dlmf.nist.gov
- NIST Digital Library of Mathematical Functions. §27.3 Multiplicative Properties (formulas 27.3.5 and 27.3.6 for d(n) and σ(n)). dlmf.nist.gov
- Euclid, Elements, Book VII, Definition 22 (perfect numbers), with commentary by D. E. Joyce, Clark University. mathcs.clarku.edu
- OEIS Foundation. A005101: Abundant numbers (definitions of abundant, perfect and deficient; first odd abundant number 945). oeis.org
Results are for educational purposes. Every worked number on this page was recomputed with the same functions the calculator uses. Methodology reviewed: 2026.