Three ways to find the greatest common factor, from listing factors to the Euclidean algorithm, plus GCF of three numbers and of monomials.
What is the greatest common factor?
The greatest common factor (GCF) of two or more numbers is the largest number that divides all of them evenly. The GCF of 18 and 24 is 6, because 6 goes into both and nothing bigger does.
You’ll also see it called the greatest common divisor (GCD) or highest common factor (HCF). They all mean the same thing. If two numbers share no factor except 1, like 15 and 28, their GCF is 1 and they are called relatively prime.
How to find GCF by listing factors
To find the GCF by listing, write out every factor of each number and pick the largest one they share. This is fastest when the numbers are under about 50.
How do you find the greatest common factor with prime factorization?
Break each number into primes, then multiply the primes they have in common, using the lowest power of each. This method scales better than listing because you never write out long factor lists.
Take 84 and 120:
What is the Euclidean algorithm for GCF?
The Euclidean algorithm finds the GCF by repeated division: divide the larger number by the smaller, replace the larger with the remainder, and repeat until the remainder is 0. The last nonzero remainder is the GCF.
It’s the best method for big numbers, because you never need to factor anything. Here it is for 1071 and 462:
How do you find the GCF of three numbers?
Factor all three into primes and keep only the primes that appear in every number, at the lowest power. A prime missing from even one number is left out.
How do you find the GCF of monomials like 12x³ and 18x²?
Find the GCF of the coefficients, then take each shared variable to its lowest power. For 12x³ and 18x², the GCF is 6x².
Here’s why. The numbers 12 and 18 have GCF 6. For the variable, x³ means x · x · x and x² means x · x. They share two x’s, so x² goes into both, and x³ does not go into x².
Which GCF method should you use?
Pick the method by the size of the numbers. All three give the same answer.
What is the GCF used for?
The GCF is used to simplify fractions and to split things into equal groups. Dividing the top and bottom of 84/120 by their GCF of 12 gives 7/10 in one step.
For grouping, say you have 56 pencils and 98 erasers and want identical kits with nothing left over. GCF(56, 98) = 14, so you can make 14 kits, each with 4 pencils and 7 erasers.
The GCF also links to the least common multiple. For two numbers, GCF × LCM equals their product: 6 × 72 = 432, and 18 × 24 = 432.
How is the GCF different from the LCM?
The GCF is the largest number that divides into all your numbers, while the LCM is the smallest number they all divide into. One looks for shared factors, the other for shared multiples.
With prime factorization, the difference comes down to one choice. The GCF takes shared primes at the lowest power. The LCM takes every prime at the highest power.
What mistakes should you watch for?
The biggest mix-up is between GCF and LCM. The GCF is never larger than the smallest number, while the LCM is never smaller than the largest. If your GCF of 18 and 24 came out as 72, you found the LCM.
With prime factors, people sometimes take the highest power instead of the lowest. That also gives the LCM. With monomials, the slip is using x³ for 12x³ and 18x², when x³ doesn’t divide x². The GCF calculator shows both the prime factor and Euclidean steps, so you can compare your work line by line.
Frequently asked questions
How do you find the GCF of three numbers?
Write each number as a product of primes and keep only the primes that all three share, at the lowest power. For 24, 36 and 60 that is 2² × 3 = 12.
What is the GCF of 18 and 24?
The GCF of 18 and 24 is 6. Their common factors are 1, 2, 3 and 6, and 6 is the largest.
What is the fastest way to find the greatest common factor?
For small numbers, list factors. For large numbers, use the Euclidean algorithm, which only needs division and never requires factoring.
Can the GCF be 1?
Yes. When two numbers share no factor other than 1, such as 15 and 28, the GCF is 1. Those numbers are called relatively prime.
What is the GCF of 12x³ and 18x²?
It is 6x². The GCF of 12 and 18 is 6, and the lowest power of x shared by both terms is x².
What is the GCF of 105 and 147?
It is 21. With the Euclidean algorithm, 147 = 1 × 105 + 42, then 105 = 2 × 42 + 21, then 42 = 2 × 21 + 0.