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Greatest Common Factor

Greatest Common Factor
GCF

The largest integer that divides evenly into each of a given set of numbers. Often abbreviated GCF or gcf. For example, 6 is the gcf of 30 and 18. Sometimes GCF is written using parentheses: (30, 18) = 6.

 

 

See also

Least common multiple, factor

Worked Example

Problem: Find the greatest common factor of 48 and 180.
Step 1: Find the prime factorization of each number.
48=24×3180=22×32×548 = 2^4 \times 3 \qquad 180 = 2^2 \times 3^2 \times 5
Step 2: Identify the prime factors that appear in both factorizations: 2 and 3.
Step 3: For each shared prime factor, take the smaller exponent. The factor 2 appears with exponent 4 in 48 and exponent 2 in 180, so take 2². The factor 3 appears with exponent 1 in 48 and exponent 2 in 180, so take 3¹.
min⁡(4,2)=2⇒22min⁡(1,2)=1⇒31\min(4,2) = 2 \quad \Rightarrow 2^2 \qquad \min(1,2) = 1 \quad \Rightarrow 3^1
Step 4: Multiply these together to get the GCF.
GCF(48,180)=22×3=4×3=12\text{GCF}(48, 180) = 2^2 \times 3 = 4 \times 3 = 12
Answer: The greatest common factor of 48 and 180 is 12.

Another Example

Problem: Find the GCF of 60 and 90 by listing factors.
Step 1: List all factors of 60.
1,2,3,4,5,6,10,12,15,20,30,601, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Step 2: List all factors of 90.
1,2,3,5,6,9,10,15,18,30,45,901, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Step 3: Identify the common factors — numbers that appear in both lists.
1,2,3,5,6,10,15,301, 2, 3, 5, 6, 10, 15, 30
Step 4: The greatest of these common factors is the GCF.
GCF(60,90)=30\text{GCF}(60, 90) = 30
Answer: The greatest common factor of 60 and 90 is 30.

Frequently Asked Questions

How do you find the GCF of three or more numbers?
Use the same methods — prime factorization or listing factors — but applied to all the numbers at once. For prime factorization, take each prime factor that appears in every number, using the smallest exponent each time. For example, GCF(12, 18, 24): 12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3. The shared primes are 2 (smallest exponent 1) and 3 (smallest exponent 1), so the GCF is 2 × 3 = 6.
What is the GCF of two numbers that have no common factor other than 1?
Their GCF is 1, and the numbers are called relatively prime (or coprime). For example, 8 and 15 share no prime factors, so GCF(8, 15) = 1.

Greatest Common Factor (GCF) vs. Least Common Multiple (LCM)

The GCF is the largest number that divides into all given numbers; the LCM is the smallest number that all given numbers divide into. They work in opposite directions. For two numbers aa and bb, these quantities are linked by the relationship GCF(a,b)×LCM(a,b)=a×b\text{GCF}(a,b) \times \text{LCM}(a,b) = a \times b. When simplifying fractions, you use the GCF. When finding a common denominator, you use the LCM.

Why It Matters

The GCF is essential for simplifying fractions: you divide both the numerator and denominator by their GCF to reduce a fraction to lowest terms. It also appears in algebra when factoring expressions — pulling out the GCF of all terms is typically the first step. Beyond the classroom, the GCF shows up in problems involving equal grouping, such as splitting items into the largest possible identical sets.

Common Mistakes

Mistake: Choosing a common factor that isn't the greatest one.
Correction: A common factor like 2 or 3 may divide both numbers, but you need the largest such factor. After finding one common factor, always check whether a bigger one exists — or use prime factorization to guarantee you get the greatest.
Mistake: Confusing GCF with LCM and using the wrong one when simplifying fractions.
Correction: To simplify a fraction, divide by the GCF of the numerator and denominator. The LCM is used for a different purpose: finding common denominators when adding or subtracting fractions.

Related Terms