The greatest common factor (GCF) of two or more numbers is the largest number that divides evenly into all of them. For example, the greatest common factor of 12 and 18 is 6, since 6 is the largest number that divides into both without a remainder. The simplest way to find it is to list the factors of each number and pick the largest one they share. For larger numbers, prime factorization or the division method works faster.
Knowing how to find greatest common factor pays off most when simplifying fractions, since dividing the numerator and denominator by their GCF is how a fraction reduces to lowest terms. This guide covers all three methods, when to use each, and how GCF relates to LCM.
What Is the Greatest Common Factor? (The Simple Version)
The greatest common factor of two numbers is the largest number that divides evenly into both. Take 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The largest shared number is 6, so the GCF of 12 and 18 is 6.
Method 1: How to Find Greatest Common Factor by Listing Factors
This is the most straightforward way to find the greatest common factor, and it works best with smaller numbers.
• Factors of 20: 1, 2, 4, 5, 10, 20
• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The largest shared factor is 10, so the GCF of 20 and 30 is 10. Once numbers get larger, listing every factor by hand takes a while, which is where the next two methods come in.
Method 2: How to Find Greatest Common Factor Using Prime Factorization
For bigger numbers, breaking each one into prime factors is usually faster than listing every factor.
Take 36 and 48.
• 36 = 2 × 2 × 3 × 3
• 48 = 2 × 2 × 2 × 2 × 3
Take each prime factor the lowest number of times it appears in either list. Both numbers share two 2s and one 3: 2 × 2 × 3 = 12. The GCF of 36 and 48 is 12.
Method 3: The Division (Euclidean) Method
This method is fastest once numbers get large, and it works cleanly for three numbers at once, answering a common question: what is the GCF of 3 numbers?
Take 48 and 72. Divide the larger by the smaller: 72 ÷ 48 leaves a remainder of 24. Divide 48 by that remainder: 48 ÷ 24 leaves a remainder of 0. Once the remainder hits 0, the last divisor used is the answer: the GCF of 48 and 72 is 24.
For three numbers, like 24, 36, and 60, find the GCF of the first two, then find the GCF of that result and the third number. The GCF of 24 and 36 is 12, and the GCF of 12 and 60 is also 12, so the greatest common factor of all three numbers is 12.
GCF vs. LCM: What's the Difference?
GCF and LCM often get confused since they use similar steps, but they answer different questions. The greatest common factor is the largest number that divides evenly into two or more numbers. LCM, or least common multiple, is the smallest number they divide into. Our guide on what LCM is and how to find it covers that side of the comparison in full.
For 12 and 18, the GCF is 6 and the LCM is 36. A handy formula links the two: the GCF times the LCM always equals the product of the original numbers. For 12 and 18, that's 6 × 36 = 216, matching 12 × 18 = 216.
Why GCF Matters for Fractions
The most common reason to find the greatest common factor is to simplify a fraction to lowest terms. Divide both the numerator and denominator by their GCF, and the fraction shrinks without changing its value. Our guide on how to simplify fractions covers that process in full.
Take 8/12. The greatest common factor of 8 and 12 is 4. Divide both numbers by 4: 8/12 becomes 2/3, fully simplified.
GCF in Real Life
The greatest common factor shows up anytime items need to split into equal groups with nothing left over. With 24 apples and 36 oranges for identical gift baskets using all the fruit, the GCF of 24 and 36, which is 12, tells you the most baskets you can make.
Arranging objects into even rows works the same way. A garden bed with 18 tomato plants and 24 pepper plants can split into 6 equal rows of each, since 6 is the GCF of 18 and 24.
When Does My Child Learn This?
Finding the greatest common factor is part of Grade 6 math under CCSS 6.NS.B.4, building on the fraction-simplifying work from Grades 4 and 5. A shaky grasp of simplifying fractions is usually the gap worth closing first. Our guides on equivalent fractions and what is a common denominator cover the related skills.
Common Mistakes When Finding the Greatest Common Factor
A few habits tend to trip people up when learning how to find greatest common factor.
• Confusing GCF with LCM, grabbing the smallest shared multiple instead of the largest shared factor
• Stopping at a shared factor that isn't the largest one, like picking 3 instead of 6 for 12 and 18
• Forgetting to use the lowest power of each prime factor in the prime factorization method
• Listing every factor by hand for large numbers instead of using prime factorization or division
Key Takeaways
• The GCF is the largest number that divides evenly into two or more numbers.
• Listing factors work for small numbers; prime factorization and division are faster for larger numbers or three numbers at once.
• GCF and LCM differ: GCF gives the largest shared factor, LCM gives the smallest shared multiple.
• Knowing how to find greatest common factor is the key step behind simplifying fractions.
• Real-life uses include splitting items into equal groups and arranging objects in even rows.
Frequently Asked Questions
What is the greatest common factor of two numbers?
It's the largest number that divides evenly into both. For 12 and 18, the GCF is 6.
How do you find the GCF using prime factorization?
Break each number into prime factors, then multiply the lowest power of each prime factor shared by both. For 36 and 48, that gives a GCF of 12.
What is the difference between GCF and LCM?
GCF is the largest number that divides into two or more numbers. LCM is the smallest number they divide into. They answer opposite questions despite similar steps.
What is the GCF of 3 numbers?
Use the division method on the first two numbers to find their GCF, then apply it again to that result and the third number.
Knowing how to find greatest common factor makes simplifying fractions faster and helps other math topics click into place.
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