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Greatest, common, factor—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 standardized exam, or looking to get ahead in your math class, this guide is for you.
Read on for clear definitions, easy-to-follow instructions, helpful examples, practice exercises, and answers to questions students often ask.
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Before we dive into the greatest common factor, let’s take a moment to remind ourselves what factors are.
A factor is a number that divides another number evenly without leaving a remainder. Or, in other words, it's a number that can be multiplied with another to get a specific product.
Let’s look at 15 as an example.
1 × 15 = 15 \( \longrightarrow \) So, 1 and 15 are factors of 15.
3 × 5 = 15 \( \longrightarrow \) So, 3 and 5 are also factors of 15.
That means the factors of 15 are 1, 3, 5, and 15.
Now that we remember what factors are, let’s talk about the greatest of them all
The greatest common factor (GCF) is the largest number that is a factor of two or more numbers. In other words, it’s the biggest number that can divide each of the given numbers evenly.
Let’s see this in action 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.
The common factors are 1, 2, 3, and 6, but the greatest one is 6.
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Now that we know what the greatest common factor is, how do we find it?
There’s more than one way! Mathematicians have come up with three different methods to determine the GCF.
Listing Factors
Prime Factorization
The Division Method
Each method has its advantages, and the best one to use depends on the numbers we’re working with.
Let’s walk through each one step by step!
Let’s start with the most straightforward way to find the greatest common factor: listing factors!
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.
Let’s confirm this with an example.
We’ll try to find the GCF of 16 and 24.
Step 1: List the factors of each number
Factors of 16: 1, 2, 4, 8, 16
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Step 2: Find the common factors
The common factors are 1, 2, 4, and 8.
The greatest of these is 8, so the GCF of 16 and 24 is 8!
Step 3: Choose the greatest common factor
The greatest of these is 8, so the GCF of 16 and 24 is 8!
Quite simple, right?
This method is great for smaller numbers since listing factors is quick and easy. But for larger numbers, it can take too long!
There are still two more methods to explore, so let’s keep working.
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.
Step 1: Write the prime factorization of each number.
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
Step 2: Find the prime factors they have in common.
Both numbers have two 2s and one 3 in common.
Step 3: Multiply the common prime factors.
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.
The division method finds the greatest common factor by repeatedly dividing numbers until no remainder is left. The last divisor before reaching zero is the GCF.
Let’s see how it works with an example!
We’ll find the GCF of 42 and 56.
Step 1: Divide the larger number (56) by the smaller number (42).
56 ÷ 42 = 1 remainder 14 (since 42 doesn’t go evenly into 56, we focus on the remainder, 14).
Step 2: Now, divide the previous divisor (42) by the remainder (14).
42 ÷ 14 = 3 remainder 0 (since there’s no remainder, 14 is the GCF!).
So, the GCF of 42 and 56 is 14!
This method is especially useful for large numbers because it quickly finds the GCF without needing to list or factorize anything.
At Mathnasium, we offer personalized K-12 math tutoring in a fun and supportive environment to help students master concepts like the GCF with confidence.
Practice makes perfect! Let’s walk through a few more examples together to make sure we understand all three methods for finding the GCF.
Let’s find the GCF of 20 and 30 using the listing factors method.
Step 1: 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
Step 2: Find the common factors.
The common factors are 1, 2, 5, and 10.
Step 3: Find the greatest one.
The GCF of 20 and 30 is 10!
Let’s find the GCF of 42 and 63 using the prime factorization method.
Step 1: Write the prime factorization of each number.
42 = 2 × 3 × 7
63 = 3 × 3 × 7
Step 2: Identify the common prime factors.
Both numbers have one 2 and two 3s in common.
Step 3: Multiply them together.
3 × 7 = 21
So, the GCF of 42 and 63 is 21!
Let’s find the GCF of 48 and 72 using the division method.
Step 1: Divide the larger number (72) by the smaller number (48).
72 ÷ 48 = 1 remainder 24
Step 2: Divide the previous divisor (48) by the remainder (24).
48 ÷ 24 = 2 remainder 0
Step 3: Since we reached 0, the last divisor is the GCF.
The GCF of 48 and 72 is 24!
Now it’s your turn! Use what you’ve learned to find the greatest common factor in these exercises.
When you’re finished, check your answers at the bottom of the guide.
Find the GCF of 18 and 30 by listing the factors.
Use prime factorization to find the GCF of 54 and 72.
Find the GCF of 56 and 98 using the division method.
Here are some of the most frequent questions we hear about the GCF at Mathnasium of South Westminster—and the answers to help you understand them better.
Nothing! 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 (largest number that divides both)
LCM = 36 (smallest number both divide into)
This means 6 fits into both numbers, while both numbers fit into 36.
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.
Mathnasium of South Westminster is a math-only learning center for K-12 students in Westminster, CO.
Using a proprietary teaching approach called the Mathnasium Method™, our specially trained math tutors offer face-to-face instruction in an engaging and fun environment to help students of all skill levels master any elementary school math topic, including the greatest common factor, usually taught in 4th, 5th, and 6th-grade math.
Each student begins their Mathnasium journey with a diagnostic assessment that allows us to understand their unique strengths and knowledge gaps. Guided by assessment-based insights, we develop personalized learning plans that will put them on the best path toward math mastery.
Whether your student is looking to catch up, keep up, or get ahead in their math class, schedule an assessment, and enroll at Mathnasium of South Westminster today!
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If you've tried our practice exercises, check your answers here.
Task 1 (Listing Factors Method): GCF(18, 30) = 6
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
GCF = 6
Task 2 (Prime Factorization Method): GCF(54, 72) = 18
54 = 2 × 3 × 3 × 3
72 = 2 × 2 × 2 × 3 × 3
Common prime factors: 2 × 3 × 3 \Rightarrow GCF = 18
Task 3 (Division Method): GCF(56, 98) = 14
98 ÷ 56 = 1 remainder 42
56 ÷ 42 = 1 remainder 14
42 ÷ 14 = 3 remainder 0 \Rightarrow GCF = 14
Got them all right? Great job! If not, review and try again—practice makes progress!
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