The greatest common factor (GCF) is the largest factor shared by two or more numbers: the largest number that divides into both without a remainder. The GCF of 18 and 24 is 6. Find it by listing factors or by multiplying the prime factors the numbers share. The GCF is how you simplify fractions in one step.
In Spanish
máximo común divisor
Also called
GCF, greatest common divisor, GCD, highest common factor, HCF
Grade level
6-8
Grade 5Grade 6Grade 7Grade 8CCSS 6.NS.B.4GED
Greatest Common Factor in a picture
Both methods agree: the largest factor on both lists is 6, and the primes the two numbers share multiply to 6. The prime-factor method is faster for big numbers.
Formal definition and notation
For whole numbersa and b, not both 0, the greatest common factor gcd (a, b) is the largest whole number d that is a factor of both a and b. Every common factor of a and b is itself a factor of the GCF. If gcd (a, b) = 1, the numbers are relatively prime. The GCF and the least common multiple are linked by gcd (a, b) × lcm(a, b) = a × b for positive a and b.
Symbol
Read it as
Meaning
GCF(18, 24) = 6
the GCF of 18 and 24 is 6
the usual school notation
gcd (a, b)
g c d of a and b
greatest common divisor, the same number; spreadsheets use GCD
Key facts
The GCF is never larger than the smallest of the numbers. If one number divides the other, it is the GCF: GCF(7, 21) = 7.
1 is a common factor of every pair, so the GCF is at least 1. Two different primes always have GCF 1.
Factoring out the GCF rewrites a sum as a product: 18 + 24 = 6(3 + 4), and 12x + 8 = 4(3x + 2).
For two positive numbers, GCF × LCM = the product of the numbers: 6 × 72 = 18 × 24 = 432.
GCF from prime factors
GCF from prime factors
GCF = product of the shared primes, each with its smaller exponent
18 = 2 × 32
prime factorization of the first number
24 = 23 × 3
prime factorization of the second number
21 × 31 = 6
shared primes 2 and 3, each to the lower power
A prime that appears in only one of the numbers (like 5 in 60) is not shared, so it is left out. If no primes are shared, the GCF is 1.
Greatest Common Factor examples, step by step
Example 1: List the factors
Warm-up
Find the GCF of 16 and 40.
Factors of 16: 1, 2, 4, 8, 16.
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40.
Common factors: 1, 2, 4 and 8. The greatest is 8.
Answer: The GCF of 16 and 40 is 8.
Example 2: Use prime factorization
Core
Find the GCF of 36 and 60.
Factor into primes: 36 = 22 × 32 and 60 = 22 × 3 × 5.
Shared primes: 2 and 3. The 5 is in 60 only, so leave it out.
Take the lower power of each: 22 (both have 22) and 31 (60 has only one 3).
Multiply: 22 × 3 = 4 × 3 = 12.
Answer: The GCF of 36 and 60 is 12.
Example 3: Identical bouquets
Test-style
A florist has 48 roses and 60 tulips. She wants to make identical bouquets that use every flower. What is the greatest number of bouquets she can make, and what goes in each?
Identical bouquets with nothing left over means the number of bouquets must divide both 48 and 60.
The greatest such number is GCF(48, 60). 48 = 24 × 3 and 60 = 22 × 3 × 5, so the GCF is 22 × 3 = 12.
Roses per bouquet: 48 ÷ 12 = 4. Tulips per bouquet: 60 ÷ 12 = 5.
Trap: the LCM, 240, would be the answer to a “when do they match up again” question, not this one.
Answer: She can make 12 bouquets, each with 4 roses and 5 tulips.
Common mistakes
✗ MistakeThe GCF of 6 and 8 is 24.
✓ InsteadThe GCF of 6 and 8 is 2. (24 is their LCM.)
The greatest common factor is a factor, not a multiple. It can never be larger than the smaller number.
✗ MistakeThe GCF of 24 and 36 is 6, because 6 divides both.
✓ Instead6 is a common factor, but 12 is the greatest: 24 = 12 × 2 and 36 = 12 × 3.
Check that the quotients have no common factor left. Here 4 and 6 still share a 2, so 6 was not the greatest.
✗ Mistake12 = 2 × 2 × 3 and 18 = 2 × 3 × 3, so the GCF is 2 × 2 × 3 × 3 = 36.
✓ InsteadUse only the primes they share: one 2 and one 3, so the GCF is 6.
Multiplying every prime gives a multiple of both numbers, which is the LCM idea. The GCF keeps only the overlap.
✗ Mistake8 and 15 have no GCF.
✓ InsteadGCF(8, 15) = 1. The numbers are relatively prime.
1 divides every whole number, so any two whole numbers have a common factor of at least 1.
A factor belongs to one number (the factors of 18). The GCF compares two or more numbers and picks the largest factor they share.
Quick check: 3 questions
Question 1 of 3
What is the GCF of 20 and 30?
Show answer and explanation
Answer: C. The common factors of 20 and 30 are 1, 2, 5 and 10, and 10 is the greatest. 5 and 2 are common factors but not the greatest; 60 is the LCM.
Question 2 of 3
What is the GCF of 9 and 16?
Show answer and explanation
Answer: A.9 = 32 and 16 = 24 share no prime, so the only common factor is 1. 3 divides 9 but not 16, 144 is the product (and the LCM), and 0 is never a GCF.
Question 3 of 3
Use the GCF to write 1824 in simplest form.
Show answer and explanation
Answer: B. GCF(18, 24) = 6, and 18 ÷ 624 ÷ 6 = 34. 912 and 68 are equal but divide by a smaller common factor, so they are not simplest. 23 is a different number.
Greatest Common Factor: frequently asked questions
How do you find the GCF of three numbers?
Use the same methods with all three. Factor each into primes and keep only the primes that appear in every number, with the lowest power. For 12, 18 and 30: 12 = 22 × 3, 18 = 2 × 32, 30 = 2 × 3 × 5. All share one 2 and one 3, so the GCF is 6. You can also find GCF(12, 18) first, then the GCF of that and 30.
What is the GCF used for?
The GCF simplifies fractions in one step: divide the top and bottom by it. It also factors expressions, as in 15x + 20 = 5(3x + 4), and it solves sharing problems where you split groups into the largest possible equal sets with nothing left over, such as packing 24 pencils and 36 erasers into identical kits.
Can the GCF be 1?
Yes. When two numbers share no prime factors, their only common factor is 1, and they are called relatively prime (or coprime). Examples: 8 and 15, 9 and 16, or any two different primes. Neither number has to be prime itself: 8 and 15 are both composite but still have GCF 1.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.NS.B.4
Cite this entry
Effortless Math Team. “Greatest Common Factor.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/greatest-common-factor/
Flashcard
TermGreatest Common FactorGCF of 18 and 24
DefinitionThe largest number that divides into both without a remainder. GCF(18, 24) = 6, so 1824 = 34.
Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5FactorA factor is a number or expression that divides another number or expression evenly, with no remainder. Factors multiply together to make a product: 4 × 6 = 24, so 4 and 6 are factors of 24. Every whole number greater than 1 has at least two factors, 1 and itself.Number System · Grade 4, Grade 5, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Least Common MultipleThe least common multiple (LCM) is the smallest positive multiple shared by two or more numbers. The LCM of 6 and 9 is 18: it is the first number on both lists of multiples, 6, 12, 18, … and 9, 18, …. The LCM of the denominators is the least common denominator for adding fractions.Number System · Grade 5, Grade 6, Grade 7DivisorA divisor is the number you divide by. In 56 ÷ 8 = 7, the divisor is 8: it tells how many equal groups to make, or how big each group is. The divisor can never be 0. A divisor of a whole number is also one of its factors, because it divides that number with no remainder.Operations · Grade 3, Grade 4, Grade 5NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 35, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4, Grade 5DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5Simplest FormSimplest form is the way of writing a fraction so its numerator and denominator have no common factor other than 1. To get there, divide the top and bottom by their greatest common factor: 68 = 6 ÷ 28 ÷ 2 = 34. The value stays the same; only the numbers get smaller. It is also called lowest terms.Number System · Grade 4, Grade 5, Grade 6FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5Prime NumberA prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 1 is not prime, and 2 is the only even prime number. Whole numbers greater than 1 that are not prime are composite.Number System · Grade 4, Grade 5, Grade 6Prime FactorizationPrime factorization is writing a whole number as a product of prime numbers only, such as 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32. Find it with a factor tree or repeated division. Every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors.Number System · Grade 4, Grade 5, Grade 6Equivalent FractionsEquivalent fractions are fractions that name the same value, like 12, 24 and 36. To make one, multiply or divide the numerator and denominator by the same nonzero number. That multiplies the fraction by 1, so its value does not change, and the fractions land on the same point of a number line.Number System · Grade 3, Grade 4, Grade 5Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8MultipleA multiple of a number is the result of multiplying that number by any whole number. The multiples of 4 are 4, 8, 12, 16, and so on, continuing without end. Every number is a multiple of itself and of 1. To test a number, divide: if the remainder is 0, it is a multiple.Number System · Grade 4, Grade 5, Grade 6