If you’ve wondered how to teach GCF and LCM in a way your students will actually get, you are not alone. Even after teaching GCF and LCM, many teachers hear things like “How do I find GCF again?” or “Is this GCF or LCM?” Many students can calculate GCF and LCM, but they struggle to understand what these concepts actually represent or when to use each.
It is very common for students to struggle with GCF and LCM, even after they have been taught it. Students hear “greatest” and automatically think the number will be greater than the original numbers. Similarly, they think that the “least” in least common multiple means the number will be less than the original numbers. The good news is that you can teach GCF and LCM in a way that will stick with students.
In this post, we’ll break down what GCF and LCM mean, when to use each one, and a few classroom-friendly methods for finding them. Over time, I’ve found that students understand these concepts much better when the focus shifts from memorizing steps to recognizing what the problem is asking. Let’s start by reviewing what GCF and LCM actually mean in math.
What Is GCF and LCM in Math?
GCF and LCM stand for Greatest Common Factor and Least Common Multiple. The GCF is the largest factor that two or more numbers share. The LCM is the smallest multiple that two or more numbers share.
GCF and LCM are covered in 6th grade, but connect to math skills in other grade levels as well. GCF is a foundational skill for factoring in higher-level algebra classes. Finding the LCM is essentially the same process as finding the least common denominator (which students should be familiar with from 4th and 5th grade).
When to Use GCF and LCM (How I Explain It to Students)
If your students ask, “How do I know whether to use GCF or LCM?”, you are not alone. Even if they can calculate the answers correctly, knowing when to use GCF and LCM is another challenge. When students get to word problems that require finding the GCF or LCM, they need to be able to decide from context which one to use. This is why I love using word problem practice that includes both GCF and LCM problems randomly mixed In, like these digital mystery sheets.
GCF problems will often include the word “greatest”, “largest”, “most”, or another similar term. When you are breaking something into even groups, you need to find the GCF.
Example GCF problem: A teacher has 24 pencils and 36 markers and wants to make identical supply kits using all of the items. What is the greatest number of kits she can make?
This is a GCF problem because we are splitting items into the largest equal groups possible.
The LCM should be used for problems that involve things repeating or lining up over time. Whenever a question asks when two things will occur at the same time, use LCM.
Example LCM problem: One bus arrives every 6 minutes, and another arrives every 8 minutes. When will they arrive at the same time again?
This is an LCM problem because we are finding the first time that repeating patterns match.
Quick Rule for Students:
If you’re making groups, think GCF.
If something is repeating or lining up, think LCM.
Once students understand when to use GCF and LCM, the next step is choosing a method to calculate them.
How to Get GCF and LCM: Methods That Actually Work in 6th Grade
There are several different methods for finding the GCF and LCM. You can choose to introduce them all to students or teach one way to get to the correct answer. No matter what method you choose, consistency is key. It will be much harder for students to master finding the GCF and LCM if they are switching between methods.
It is also important to keep in mind that we, as teachers, should focus on students’ understanding of the concept of factors and multiples, rather than focusing on the methods. Emphasize what the greatest common factor and least common multiple are conceptually, and provide the methods for students as a way to get there.
Ladder Method for GCF and LCM
The ladder method for GCF and LCM is a fan favorite because it is good for visual learners. It is a simple algorithm that has a way of sticking with kids in a way they will remember. With this method, students don’t have to find the largest factor immediately; they can work up to it by dividing by smaller numbers. This is great for students whose knowledge of higher multiplication facts is not automatic. The ladder method is also very similar to the long-division algorithm, which students should be familiar with by 6th grade.
How to Use the Ladder Method for GCF and LCM
To use the ladder method for GCF and LCM (also known as the “grid method”), first make a grid and place the starting numbers in the top row of each column. Next, find any factor the two numbers share (it doesn’t have to be the largest, but it can be). Then, write this number in the furthest left column. Divide each original number by this factor and write the result under each original number. If these numbers still have a common factor, repeat the process by dividing again until the numbers are relatively prime. The GCF is found by multiplying all of the factors in the left column. The LCM is found by multiplying all of the numbers in the “L” shape (the left column in the bottom row).
It is very easy for students to organize the factors with this method. This can even be used to find the GCF and LCM of more than two numbers. The biggest drawback is that students often don’t understand “what” they are doing mathematically with this method. They use it to find the right answer, but it doesn’t necessarily reinforce the concept of factors and multiples. Also, watch out for students who try to multiply before they reach numbers with no more common factors.
How to Find GCF and LCM Using Prime Factorization
The Common Core State Standards do not explicitly require prime factorization, but it is a common way teachers use for how to teach GCF and LCM. The first step is to complete the prime factorization for each of your numbers by using a factor tree. Write the number at the top. Then, find any two factors of that number and write them underneath with two branches connecting them. If either number is prime, circle it. If they are not prime, continue finding the factor pairs and creating branches until you are left with only prime numbers. (see the example image below)
The GCF is found by multiplying any prime factors that appear in each factor tree. The LCM is found by multiplying the common factors by all of the remaining prime factors.
Students can often mess up the GCF by multiplying more than just the overlapping factors. It is also easy to miss a factor that isn’t written at the very bottom. Teacher tip: To reduce this error, tell students to bring the prime factors all the way to the bottom so they are not missed.
I find it particularly difficult to find the LCM this way, because it can be easy to multiply a common factor more than once. It is a very common mistake for students to multiply all of the prime factors they see, which will result in an incorrect answer. I find this method to be too confusing
How to Find GCF and LCM Using Factor Rainbows and the List Method
The method I really like for how to teach GCF and LCM is by using factor rainbows and the list method. It reinforces the concepts of factors and multiples and is the simplest method to use. Another plus is that students must actually understand what factors and multiples are to find the answers.
To find the GCF using a factor rainbow, start with the original number and test each descending number for divisibility. List each factor with its pair. Then choose the largest number that appears in both lists. This works better for smaller numbers, although it can work for large numbers that have a short list of factors.
To find the LCM using the list method, list the first few multiples of each number until you reach the first one that is shared. That’s it! This works best for numbers that have a relatively small LCM. Otherwise, students could end up listing multiples for a while.
This method is great for students who have trouble remembering the steps of other methods. It is also usually the quickest method for finding GCF and LCM. The list method is particularly good for students who struggle with multiplication because they can simply perform repeated addition to get the multiples. However, the factor rainbow requires students to have a solid foundation of their multiplication facts, and factor pairs can be missed with this method.
There is no fancy trick with this method, so sometimes students forget what to do. But I find that the way this method reinforces the concepts of factors and multiples is worth it. It clearly shows that factors are smaller than their multiples and that while factors are limited, multiples can go on forever.
Why Students Still Mix Up GCF and LCM (Even After You Teach It)
Even after students learn how to calculate the greatest common factor and the least common multiple, many still struggle to decide which one to use in a problem. The issue usually isn’t the math—it’s the context.
For starters, the vocabulary sounds similar to students. They hear “greatest”, “least”, “factor”, and “multiple”, and these words sound abstract and interchangeable to them. Students may more easily remember terms like “ladder method”, but the terms “factor” and “multiple” don’t always stick. I’ve found that one key to student success is making sure students are solid on the definitions of factors and multiples (math vocabulary is so important to overall understanding!)
Over-procedural teaching can be another roadblock to student success. Sometimes students are taught how to find GCF and LCM, but not why each one is used. When instruction focuses mostly on the steps, the students don’t know why they are doing what they are doing. To fully master GCF and LCM, students need to understand what it is they are finding.
Many worksheets may include good GCF and LCM problems, but they don’t give students feedback when they are wrong. As a teacher, we cannot be with all students at the same time, so some are completing multiple problems in a row incorrectly before they get feedback. The more an error is repeated, the more it is solidified in a student’s brain. That’s why giving students meaningful practice that reinforces the difference between these concepts can make a big impact.
An Engaging Way to Reinforce GCF and LCM
At the end of the day, practice makes perfect. Repetition and review are the keys to reinforcing GCF and LCM skills. This is why I like using a self-checking activity like these digital GCF and LCM mystery pictures to help students practice once they know the basics. Students will immediately know whether the answers they type in are right or wrong. This immediate feedback is perfect for catching mistakes early. It also helps students evaluate if the method they are using is working for them. The pixel art pictures that slowly reveal after each correct answer are also a great way to keep practice engaging. We all know that completing multiple math problems of the same kind in a row is not the most exciting task for kids. Check them out here if you are interested in a ready-to-use GCF and LCM practice – complete with word problems!
Related Articles:
- GCF and LCM (Explained w/ 7 Step-by-Step Examples!)
- GCF And LCM – Math Steps, Examples & Questions
- The Cake Method for GCF and LCM
- Read my blog post on how to strengthen math vocabulary with I Have Who Has Games
Teaching GCF and LCM can be tricky, and initial confusion is normal for students. When students learn to recognize whether a situation involves grouping items or repeating events, it becomes much easier to decide whether to use the greatest common factor or the least common multiple. Students can become much more confident with these concepts when we provide instruction that focuses on a conceptual understanding and the right kind of practice. With clear explanations, meaningful practice, and engaging review activities, you’ll have a strong approach for how to teach GCF and LCM, so students truly understand the difference between them.