‘’Greatest’’, ‘’common’’, and ‘’factor’’ are three words you’ll hear a lot in math. We first learn about them in elementary school, but they stay with us throughout our math journey.
Whether you're just starting to learn about the greatest common factor, getting ready for a test, or looking to get ahead in your math class, this guide is for you.
Today, Mathnasium tutors walk you through what the greatest common factor is, three methods on how to find it, solved examples, practice problems, and answers to questions students frequently ask.
What is the greatest common factor? The largest number that divides two or more numbers evenly.
What does GCF stand for? Greatest Common Factor. It's also called the Highest Common Factor (HCF).
How do you find the GCF? There are three methods: Listing Factors, Prime Factorization, and the Division Method.
Can the GCF be larger than the numbers you're working with? No. The GCF is always less than or equal to the smallest number.
Can two numbers have a GCF of 1? Yes. When that happens, the numbers are called relatively prime or coprime.
Where does the GCF show up in math? Simplifying fractions, solving ratio problems, and working with algebraic expressions.
A factor is a whole number that divides another number evenly without leaving a remainder. Let’s put it another way, it's a number that multiplies by another number to get a specific product.
Take 15 as an example.
1 × 15 = 15
This tells us that 1 and 15 are factors of 15.
3 × 5 = 15
3 and 5 are also factors of 15.
That means the factors of 15 are 1, 3, 5, and 15.
Now, if we change the focus from multiplication to division, let’s divide 15 by all its factors, and confirm that there are no remainders.
15 ÷ 1 = 15
15 ÷ 15 = 1
15 ÷ 3 = 5
15 ÷ 5 = 3
Either way, whether through multiplication or division, the result tells us the same thing: 1, 3, 5, and 15 are all factors of 15.
📕 You May Also Like: What Is Factoring in Math? A Beginner’s Guide
The greatest common factor (GCF) is the largest number that divides two or more numbers evenly. To find it, we look at the factors two numbers share and identify the largest one.
Here's how that works with 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.
We can tell that numbers 12 and 18 share the factors 1, 2, 3, and 6, and the greatest factor is 6.
There are three methods for finding the GCF, and the best one depends on the numbers we are working with:
Listing Factors
Prime Factorization
The Division Method
Let’s walk through each one step by step.
This method is exactly what it sounds like. We list all the factors of each number and find the biggest one they have in common. Here's how it works with 16 and 24.
Factors of 16 are 1, 2, 4, 8, 16. For 24, factors are 1, 2, 3, 4, 6, 8, 12, 24.
The common factors for both 16 and 24 are 1, 2, 4, and 8.
The greatest factor from the list of common factors (1, 2, 4, 8) for 16 and 24 is 8, so the GCF is 8.
This method is great for smaller numbers since listing factors is quick and easy, but with larger ones, listing every factor takes too long. That's when the next two methods come in handy.
📕 You May Also Like: How to Find All the Factors of Any Number
The prime factorization method, as the name suggests, helps us find the greatest common factor by breaking numbers down into prime factors.
In case you've forgotten, a prime number is a number that can only be divided by 1 and itself (like 2, 3, 5, 7, 11, etc.). A prime factor is a prime number that multiplies with others to make a bigger number.
Let's find the GCF for 36 and 48 using the prime factorization method.
To do the prime factorization of any number, we need to start by dividing it by the smallest prime number (2). Let’s start with 36.
36 ÷ 2 = 18
Then, we need to check if 18 can be divided by the same prime number (2). Yes, we can.
18 ÷ 2 = 9
We cannot divide 9 by 2, so we move to the next prime number, which is 3.
9 ÷ 3 = 3
Number 3 can be divided by the same prime number once again (3).
3 ÷ 3 = 1
Now, we have reached 1, and the prime factorization ends here. So, the prime factors are the numbers we divided by (2, 2, 3, 3).
For number 48, the prime factorization looks like this:
48 ÷ 2 = 24
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
3 ÷ 3 = 1
We successfully found prime factors for 48, which are 2, 2, 2, 2, 3.
If we compare both lists (2, 2, 3, 3 and 2, 2, 2, 2, 3), we see that both numbers have two 2s and one 3 in common.
Now, we only need to multiply the numbers in common (2, 2, 3).
2 × 2 × 3 = 12.
So, the GCF of 36 and 48 is 12.
This method is great for larger numbers because it helps us find the GCF without listing every factor.
📕 You May Also Like: What Is Prime Factorization? Explain It to a 10-Year-Old
To use the division method, we repeatedly divide two numbers until there's no remainder left. The last divisor we use is the GCF. Here's how it works with 42 and 56.
Let’s start by dividing 56 by 42. Our result is 1, and the remainder is 14. Since 42 doesn't divide evenly into 56, we move to the next step using the remainder, 14.
Now, let’s divide 42 by the remainder (14). Our result is 3, and the remainder is 0 this time around. Since there's no remainder, we're done.
Since we did the last division by 14, the GCF of 42 and 56 is 14.
This method is the most efficient of the three, especially useful when we're working with large numbers and want to skip listing factors or building prime factorizations altogether.
Let’s go through a few solved examples together!
Let’s find the GCF of 20 and 30 by using the listing factors method.
First, let’s list the factors of each number.
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Now, let’s compare the lists. We can see that the common factors of 20 and 30 are 1, 2, 5, and 10.
Finally, we need to choose the greatest common factor, which is 10. That’s the GCF of 20 and 30
For this example, let’s use the prime factorization Method to find the GCF of 42 and 63.
Let’s write the prime factorization of each number first.
42 = 2 × 3 × 7
63 = 3 × 3 × 7
Then, we need to identify the common prime factors. Both numbers share one 3 and one 7.
Finally, we multiply the common prime factors. 3 × 7 = 21. The GCF of 42 and 63 is 21.
For these numbers (48,72), let’s find the GCF using the division method.
First, we divide the larger number by the smaller number.
72 ÷ 48 = 1, remainder 24
Then, let’s divide the previous divisor (48) by the remainder (24). The result is 2, and the remainder is 0. Since there's no remainder, we're done.
The last divisor (24) is the GCF.
Ready to practice what we’ve covered? Try these practice problems on your own and check your answers at the bottom of the guide.
Find the GCF of 18 and 30 using the listing factors method.
Find the GCF of 54 and 72 using the prime factorization method.
Find the GCF of 56 and 98 using the division method.
We know the greatest common factor can bring up a few questions, so we’ve put together clear answers to the ones students ask most often.
There is no difference. The greatest common factor (GCF) and the highest common factor (HCF) are just two names for the same thing, the largest number that divides two or more numbers evenly.
The GCF (greatest common factor) is the largest number that evenly divides two or more numbers. The LCM (least common multiple) is the smallest number that both numbers divide into evenly.
For example, with 12 and 18:
GCF = 6 (the largest number that divides both)
LCM = 36 (the smallest number that both divide into)
This means 6 fits into both numbers, while both numbers fit into 36.
📕 You May Also Like: What Is the Least Common Multiple? A Kid-Friendly Guide
Yes! When two numbers have no common factors other than 1, they are called relatively prime or coprime. For example, 8 and 15 have a GCF of 1 because they share no other factors.
No, the GCF can never be larger than the smallest number. The greatest common factor is always less than or equal to the smallest number given.
The process is the same! You find the common factors for all the numbers and pick the largest one they share.
The GCF shows up in many areas of math! We use it to simplify fractions, solve ratio problems, and work with algebraic expressions.
For example, in fractions, dividing the numerator and denominator by the GCF helps reduce them to their simplest form. In ratios, the GCF helps scale numbers down while keeping the same proportion.
At Mathnasium, specially trained tutors help students build a deep understanding of what the GCF is and how to find it, one step at a time.
Mathnasium is a math-only learning center dedicated to helping K–12 students of all skill levels learn and master math.
Each student starts their Mathnasium enrollment with a diagnostic assessment that helps us identify their current skills, knowledge gaps, and learning goals. From there, we build a personalized learning plan tailored to their needs and pace.
Our specially trained tutors use the Mathnasium Method™, a proprietary teaching approach that combines verbal, visual, mental, tactile, and written techniques to help students understand the math they are working with.
If students get stuck on a concept like factoring and the GCF, we break it down into manageable steps and teach both the how and the why behind it.
Gradually, students learn to do the same independently and walk out of our centers with the problem-solving skills and critical thinking tools they can use in math and beyond.
Fun is an important part of how we work. Sessions often include game-based and hands-on activities that keep students engaged and learning enjoyable. Students earn rewards along the way, and every bit of progress gets celebrated, so confidence grows alongside mastery.
Our method brings measurable results:
94% of parents report an improvement in their child's math skills and understanding
93% of parents report their child's improved attitude toward math after attending Mathnasium
90% of students saw an improvement in their school grades
With over 1,100 centers, Mathnasium brings top-rated instruction close to your home.
For families in Westminster, Mathnasium of South Westminster is a trusted center with years of experience transforming how children think and feel about math.
Here is what one parent had to say about their child's experience at Mathnasium:
Whether your child needs help catching up, wants to stay on track, or is ready to move ahead, Mathnasium can support their journey.
Ready to get started?
📅 Schedule a Free Diagnostic Assessment at Mathnasium of South Westminster
Not near South Westminster?
📍 Find Mathnasium Learning Centers Near You
Great job working through the practice problems! Here are the answers:
1. GCF(18, 30) - Listing Factors Method
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The GCF is 6 because it’s the greatest number that both 18 and 30 share.
2. GCF(54, 72) - Prime Factorization Method
54=2333
72=22233
Both 54 and 72 share one 2 and two 3s. So, we multiply 233, and the GCF is 18.
3. GCF(56, 98) - Division Method
9856=1, remainder 42
5642=1, remainder 14
4214=3, remainder 0
The GCF of 56 and 98 is 14, since that is the last number we divided by.
How did you do?
Mathnasium of South Westminster is a math-only learning center for K-12 students in Westminster, CO. Trusted by over a million parents, Mathnasium uses personalized learning plans and the proprietary Mathnasium Method™ to help students catch up, keep up, and get ahead on their math journey.
Our specially trained tutors deliver face-to-face instruction in a supportive and fun small-group environment, working with students both in center and online to develop a deep understanding of math, build confidence, and improve academic performance.
Schedule Free Assessment
(720) 524-4533
(720) 524-4533
