A common factor calculator lists every whole number that divides two numbers exactly and identifies the greatest common factor (GCF). Formula: g = gcd(a,b). Common factors = every positive divisor of g. Headline lists those factors; detail also shows GCF. The common factors of 36 and 60 are 1, 2, 3, 4, 6 and 12, so the GCF is 12.
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How to use the common factor calculator
- Enter two positive whole numbers.
- Read the list of common factors and the GCF.
How the common factor calculator works
A factor of a number divides it with no remainder. The common factors of two numbers are the factors they share, and the largest is the greatest common factor (GCF). The common factor calculator finds the GCF with Euclid’s algorithm and then lists every factor of the GCF, because the common factors of two numbers are exactly the factors of their GCF.
- Euclid’s algorithm: replace the larger number with the remainder of dividing it by the smaller, and repeat until the remainder is 0. The last non-zero number is the GCF.
- Example: 60 ÷ 36 leaves 24; 36 ÷ 24 leaves 12; 24 ÷ 12 leaves 0, so the GCF is 12.
Common factor calculator worked example
| Numbers | Common factors | GCF |
|---|---|---|
| 36, 60 | 1, 2, 3, 4, 6, 12 | 12 |
| 12, 18 | 1, 2, 3, 6 | 6 |
| 24, 40 | 1, 2, 4, 8 | 8 |
| 45, 75 | 1, 3, 5, 15 | 15 |
| 14, 25 | 1 | 1 |
| 100, 150 | 1, 2, 5, 10, 25, 50 | 50 |
For 36 and 60, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36, and the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. The numbers on both lists are 1, 2, 3, 4, 6 and 12, so the GCF is 12. When the only common factor is 1, as for 14 and 25, the numbers are called coprime.
Using prime factors
Another method is prime factorization. 36 = 2² × 3² and 60 = 2² × 3 × 5. Take each prime the two numbers share, at the lower power: 2² × 3 = 12. This method is handy for checking by hand and shows why every common factor divides the GCF.
Where common factors are used
- Simplifying fractions: divide the top and bottom by the GCF. 36/60 becomes 3/5.
- Equal groups: with 36 apples and 60 oranges, the largest number of identical gift bags with nothing left over is 12, each holding 3 apples and 5 oranges.
- Tiling and cutting: the largest square tile that fits a 36 × 60 cm area exactly is 12 cm.
- Ratios: 36:60 simplifies to 3:5.
GCF and LCM
The least common multiple (LCM) is the smallest number both numbers divide into. For two positive whole numbers, GCF × LCM = the product of the numbers. Since 36 × 60 = 2,160 and the GCF is 12, the LCM is 2,160 ÷ 12 = 180. Use the GCF to simplify and the LCM to find common denominators.
Inputs
The common factor calculator takes two positive whole numbers. For three or more numbers, find the GCF of the first two, then the GCF of that result and the next number, and so on.
Listing factors by hand
To list the factors of a number by hand, test divisors from 1 upward and stop at its square root, because factors come in pairs. For 60, the square root is about 7.7, so test 1 to 7: 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12 and 6 × 10. Those six pairs give all twelve factors. Doing the same for 36 (1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6) and comparing the two lists gives the common factors. A common factor calculator does this instantly, but the pairing method is a good way to check a result in class.
Divisibility rules speed up the testing, and they are also how a common factor calculator can be checked quickly. 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 9 if its digits add up to a multiple of 9. Since 3 + 6 = 9, 36 is divisible by both 3 and 9; since 6 + 0 = 6, 60 is divisible by 3 but not by 9, which is why 9 is a factor of 36 but not a common factor.
Related converters and calculators
Related calculators:
- GCF calculator: GCF
- LCM calculator: LCM
- Prime factorization calculator: primes
- Factor calculator: factors
Sources
OpenStax Prealgebra 2e, 2.4: Find multiples and factors; OpenStax Prealgebra 2e, 2.5: Prime factorization and the least common multiple.
Formula and method
g = gcd(a,b). Common factors = every positive divisor of g. Headline lists those factors; detail also shows GCF.
Common factors are exactly the divisors of the greatest common factor. Positive integers only.
Questions about this calculator
What are the common factors of 36 and 60?
1, 2, 3, 4, 6 and 12.
What is the GCF?
The largest common factor.
What does coprime mean?
The only common factor is 1.
How are GCF and LCM related?
GCF × LCM = product of the two numbers.
Can I use three numbers?
Find the GCF of two, then with the next.
Check the assumptions and units before relying on a result. Financial, tax and health estimates do not replace advice for your individual circumstances.
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