Common Factor Calculator
List every common factor of two or more numbers, with the greatest common factor, factor pairs and the working.
Each of these divides 24, 36 and 60 with no remainder.
Greatest common factor
Every common factor divides 12, and every factor of 12 is a common factor.
Common factor pairs that multiply to 12:
- 1 × 12
- 2 × 6
- 3 × 4
Your calculation
The same steps the calculator runs, with your numbers.
- Prime factors: 24 = 2³ × 3, 36 = 2² × 3², 60 = 2² × 3 × 5.
- Keep each prime found in every number at its lowest power: 2² × 3 = 12, the greatest common factor.
- Count the common factors from the exponents of 12: (2 + 1) × (1 + 1) = 6.
- List the factors of 12 in pairs: 1 × 12, 2 × 6, 3 × 4.
- Check: 24 ÷ 12 = 2, 36 ÷ 12 = 3, 60 ÷ 12 = 5, all whole numbers.
Factors of each number
Highlighted factors are shared by every number.
| Number | Factors | Count |
|---|---|---|
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 9 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 12 |
Common factors are positive whole numbers. A negative input is treated by its absolute value, since −24 and 24 have the same positive factors.
What Is a Common Factor?
A common factor of two or more whole numbers is a number that divides each of them with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12, and the factors of 18 are 1, 2, 3, 6, 9 and 18. The numbers in both lists, 1, 2, 3 and 6, are the common factors of 12 and 18.
The largest common factor is the greatest common factor (GCF), also called the greatest common divisor (GCD) or highest common factor (HCF). For 12 and 18 it is 6. Every common factor divides the GCF, and every factor of the GCF is a common factor, so listing the factors of the GCF lists all common factors at once.
This common factor calculator accepts 2 to 10 whole numbers up to one trillion. It lists every shared factor, shows the GCF in prime form, pairs the common factors, adds them up, gives the least common multiple and highlights the shared factors in a full factor table for each number.
- 1 is always a common factor, because 1 divides every whole number.
- No common factor exceeds the smallest number, because a factor of a positive number is never larger than the number.
- Numbers whose only common factor is 1 are coprime, also called relatively prime.
How to Find Common Factors: Three Methods
1. List the factors and compare
Write every factor of each number, then keep the ones that appear in every list. To list factors without missing any, test divisors in pairs from 1 upward and stop once the divisor passes the square root. For 18: 1 × 18, 2 × 9, 3 × 6, and 4 and 5 do not divide 18. This works well for small numbers.
2. Use prime factorization
Factor each number into primes, keep every prime that appears in all of them at its lowest power, and multiply. That product is the GCF, and its factors are the common factors.
3. Count and add the common factors without listing them
If the GCF is pa × qb × …, the number of its factors, and so the number of common factors, is (a + 1)(b + 1)…. The sum of the common factors is the product of (1 + p + … + pa) over each prime. These are the standard divisor count and divisor sum formulas.
For large numbers, find the GCF with the Euclidean algorithm instead of factoring. For 1,071 and 462: 1,071 = 2 × 462 + 147, then 462 = 3 × 147 + 21, then 147 = 7 × 21 + 0, so the GCF is 21 = 3 × 7 and the common factors are 1, 3, 7 and 21.
Common Factors Chart
Common factors of number pairs that come up often in homework and fraction problems, computed with this calculator's code.
| Numbers | Common factors | GCF | Count |
|---|---|---|---|
| 8 and 12 | 1, 2, 4 | 4 | 3 |
| 9 and 15 | 1, 3 | 3 | 2 |
| 12 and 18 | 1, 2, 3, 6 | 6 | 4 |
| 14 and 21 | 1, 7 | 7 | 2 |
| 15 and 25 | 1, 5 | 5 | 2 |
| 16 and 24 | 1, 2, 4, 8 | 8 | 4 |
| 18 and 27 | 1, 3, 9 | 9 | 3 |
| 20 and 30 | 1, 2, 5, 10 | 10 | 4 |
| 24 and 36 | 1, 2, 3, 4, 6, 12 | 12 | 6 |
| 30 and 45 | 1, 3, 5, 15 | 15 | 4 |
| 36 and 48 | 1, 2, 3, 4, 6, 12 | 12 | 6 |
| 42 and 56 | 1, 2, 7, 14 | 14 | 4 |
| 45 and 60 | 1, 3, 5, 15 | 15 | 4 |
| 54 and 72 | 1, 2, 3, 6, 9, 18 | 18 | 6 |
| 60 and 90 | 1, 2, 3, 5, 6, 10, 15, 30 | 30 | 8 |
| 72 and 108 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 36 | 9 |
| 120 and 180 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 60 | 12 |
How to Use the Calculator and Read the Result
- Enter the numbers. Type 2 to 10 whole numbers separated by commas, spaces or new lines. Each can be as large as 1,000,000,000,000. Commas inside a number, such as 1,071, are read as separators, so type large numbers without thousands separators.
- Read the list at the top. The large line shows every common factor in order, and the chip counts them. When there are more than 12, the top shows the count and the full list moves into the card below it.
- Use the greatest common factor card. It shows the GCF in prime form and the common factor pairs, which are the pairs of common factors that multiply to the GCF.
- Check the tiles and table. The tiles give the common primes, the sum of the common factors and the least common multiple. The table lists all factors of each number with the shared ones highlighted, so you can see why a factor such as 5 drops out.
A result of only 1 means the numbers are coprime. That is a valid answer, not an error: 15 and 28 share no prime, so 1 is their only common factor.
Where Common Factors Are Used
Simplifying fractions
Dividing the numerator and denominator by any common factor gives an equivalent fraction. The common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12. Dividing 24/36 by 2 gives 12/18, which still has common factors, while dividing by the GCF, 12, gives 2/3 in one step. A fraction is in simplest form when the only common factor left is 1.
Splitting items into equal groups
Suppose 24 pencils and 36 erasers go into identical kits with nothing left over. The number of kits must divide both 24 and 36, so it can be any common factor: 1, 2, 3, 4, 6 or 12 kits. The largest choice, 12 kits, puts 2 pencils and 3 erasers in each.
Cutting and tiling with no waste
To cut a 48 cm by 180 cm board into equal squares with nothing left over, the side of each square must be a common factor of 48 and 180. The common factors are 1, 2, 3, 4, 6 and 12 cm. The largest square, 12 cm, gives 48 ÷ 12 = 4 squares across and 180 ÷ 12 = 15 down, or 60 squares.
Factoring algebra expressions
Algebra uses the same idea with terms. The coefficients of 6x + 9 have common factors 1 and 3, so 6x + 9 = 3(2x + 3). Factoring out the greatest common factor is the first step in most factoring problems.
Common Mistakes
- Mixing up factors and multiples. Factors of 12 divide 12 (1, 2, 3, 4, 6, 12). Multiples of 12 are divisible by 12 (12, 24, 36 and so on). Common factors are never larger than the smallest number.
- Keeping a factor shared by only some numbers. For 24, 36 and 60, the factor 5 divides 60 only, so 5, 10 and 15 are not common factors of the set.
- Leaving out 1 or the GCF itself. Both belong in the list. For 12 and 18, the list is 1, 2, 3 and 6, not just 2 and 3.
- Stopping the factor search too early. Test divisors up to the square root and record both numbers of each pair, or the larger factors get lost.
- Thinking coprime means prime. 8 and 9 are coprime, yet neither is prime.
Common Factors, GCF, LCM and Coprime Numbers
| Term | Meaning | 24 and 36 |
|---|---|---|
| Common factor | Divides every number | 1, 2, 3, 4, 6, 12 |
| GCF | Largest common factor | 12 |
| Common multiple | Divisible by every number | 72, 144, 216, … |
| LCM | Smallest common multiple | 72 |
| Coprime | Only common factor is 1 | No (24 and 35 are) |
For two numbers, GCF × LCM equals the product of the numbers: 12 × 72 = 864 = 24 × 36. This rule does not extend to three or more numbers.
Frequently Asked Questions
What are the common factors of 12 and 18?
The common factors of 12 and 18 are 1, 2, 3 and 6. The factors of 12 are 1, 2, 3, 4, 6 and 12, and the factors of 18 are 1, 2, 3, 6, 9 and 18. The numbers that appear in both lists are the common factors. The largest of them, 6, is the greatest common factor, and every other common factor divides 6. Because 6 = 2 × 3, the count of common factors is (1 + 1) × (1 + 1) = 4, which matches the list.
How do you find the common factors of two numbers?
Find the greatest common factor first, then list its factors. For 48 and 180, the prime factorizations are 48 = 2⁴ × 3 and 180 = 2² × 3² × 5. Keep each prime that appears in both with its lower power: 2² × 3 = 12. The factors of 12 are 1, 2, 3, 4, 6 and 12, and those are exactly the common factors of 48 and 180. Listing every factor of both numbers and picking the matches gives the same answer but takes longer for large numbers.
How do you find common factors of three numbers?
Use the same method with all three numbers. A common factor must divide every number in the set, not just two of them. For 24, 36 and 60: 24 = 2³ × 3, 36 = 2² × 3² and 60 = 2² × 3 × 5. The primes found in all three are 2, with lowest power 2², and 3, with lowest power 3¹, so the greatest common factor is 12 and the common factors are 1, 2, 3, 4, 6 and 12. The prime 5 divides only 60, so 5, 10 and 15 are not common factors.
What is the difference between a common factor and the greatest common factor?
A common factor is any whole number that divides every number in a set with no remainder. The greatest common factor (GCF, also called the GCD or HCF) is the largest of them. 24 and 36 have six common factors, 1, 2, 3, 4, 6 and 12, and the greatest is 12. Every common factor is a factor of the GCF, so the GCF holds all the information: list its factors and you have every common factor. Use the GCF to simplify a fraction in one step.
What does it mean if the only common factor is 1?
The numbers are called coprime, or relatively prime. They share no prime factor, and their greatest common factor is 1. 15 and 28 are coprime: 15 = 3 × 5 and 28 = 2² × 7 have no prime in common. Coprime numbers do not have to be prime themselves, and 8 and 9 are coprime even though both are composite. A fraction whose numerator and denominator are coprime, such as 15/28, is already in simplest form, and the least common multiple of two coprime numbers is their product, here 420.
Is 1 a common factor of every set of numbers?
Yes. 1 divides every whole number, so it is a common factor of any set of whole numbers and always appears first in the list. The greatest common factor can be no bigger than the smallest number in the set, because a factor of a positive number cannot exceed it. So every set of positive whole numbers has at least one common factor. For 32 and 48, the five common factors 1, 2, 4, 8 and 16 are all the factors of 32 except 32 itself, because 32 does not divide 48.
Sources & References
- OpenStax. Prealgebra 2e, 2.4 Find Multiples and Factors (factors, multiples, finding all factors). openstax.org
- OpenStax. Elementary Algebra 2e, 7.1 Greatest Common Factor and Factor by Grouping (GCF by prime factors, factoring out the GCF). openstax.org
- OpenStax. Prealgebra 2e, 4.2 Multiply and Divide Fractions (simplified form of a fraction). openstax.org
- NIST Digital Library of Mathematical Functions. 27.3 Multiplicative Properties (divisor count and divisor sum formulas). dlmf.nist.gov
- Paul E. Black, NIST. Euclid's algorithm, Dictionary of Algorithms and Data Structures. xlinux.nist.gov
Results are for educational purposes. Every number on this page was recomputed with the same code the calculator uses. Methodology reviewed: 2026.