Just like the multiplication table is the building block of so much math, the least common multiple (LCM) has its own starring role.
From simplifying fractions to syncing schedules, the LCM weaves through different math problems.
If you’re just learning about the least common multiple, need a refresher, or are prepping for an exam, you’re in the right place.
This simple, student-friendly guide has everything you need to know—easy definitions, step-by-step methods, worked examples, practice exercises, and answers to common questions.
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Before we talk about the least common multiple, let’s make sure we understand what a multiple is.
A multiple of a number is what you get when you multiply that number by whole numbers (1, 2, 3, 4, and so on).
For example:
The multiples of 3 are: 3, 6, 9, 12, 15, 18, …
The multiples of 5 are: 5, 10, 15, 20, 25, 30, …
Since multiplication never stops, multiples go on forever!
What Is the Least Common Multiple?
The least common multiple (LCM) of two or more numbers is the smallest multiple they have in common.
Let’s see this in action with 4 and 6:
The multiples of 4 are: 4, 8, 12, 16, 20, …
The multiples of 6 are: 6, 12, 18, 24, 30, …
The smallest number that appears in both lists is 12, so the LCM of 4 and 6 is 12.
Here’s another way to think about it:
Imagine two traffic lights: one changes every 30 seconds, and the other every 45 seconds.
At first, they switch at different times, but if you wait long enough, they’ll eventually turn green together.
The lights turn green together at 90 seconds—the first moment their patterns match up, just like the LCM! By then, the first light will have switched three times (30 × 3 = 90), and the second light will have switched twice (45 × 2 = 90).
Now that we’re clear on what the least common multiple is, let’s see how to find it. There are three main methods we can use:
We’ll go through each one step by step so you can choose whichever works best for you!
The listing multiples method is the simplest way to find the least common multiple. It works just like it sounds—you list out the multiples of each number and look for the smallest one they share.
Let’s see how it works with an example!
Using this method, we’ll find the LCM of 6 and 8:
So, the LCM of 6 and 8 is 24!
Pretty simple, right?
Now, while finding the LCM by listing multiples is quick and easy—and helps visualize everything clearly—it works best with smaller numbers.
However, if you're working with larger numbers like 72 or 124, listing all the multiples can take a while.
But don’t worry, this is just the first method—there are still two more to explore.
The prime factorization method helps us find the least common multiple by breaking numbers down into their prime factors.
And do we remember what prime factors are? Let’s remind ourselves just in case.
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.
So, for this method, instead of listing out multiples, we use prime numbers to build each value from the ground up.
Let’s look at an example to see how this works.
We’ll find the LCM of 48 and 72 using prime factorization:
Break each number into prime factors:
48 = 2 × 2 × 2 × 2 × 3 or 2⁴ × 3
72 = 2 × 2 × 2 × 3 × 3 or 2³ × 3²
Take the highest power of each prime:
The prime factors we find in 48 and 72 are 2 and 3.
The highest power of 2 is 2⁴ (from 48).
The highest power of 3 is 3² (from 72).
Multiply the selected prime factors together:
2⁴ × 3² = 16 × 9 = 144
So, the LCM of 48 and 72 is 144!
This method works well for bigger numbers because we don’t have to list out long rows of multiples.
Check out our video guide on prime factorization:
The division method, also called the ladder method, is another way to find the LCM where we divide both numbers at the same time using their common factors.
In case you have forgotten, a common factor is a number that divides both numbers evenly. For example, the common factors of 12 and 18 are 1, 2, 3, and 6.
Let’s see how this method would work for finding the LCM for
Write the number side by side
We start by writing 54 and 90 next to each other like so:
2. Keep dividing by the smallest common factor until none are left
The smallest number that divides both 54 and 90 is 2, so we divide and write the results below:
Now, both 27 and 45 are divisible by 3, so we divide by 3 next:
Again, both 9 and 15 are divisible by 3, so we divide by 3 again:
Finally, 3 and 5 have no common factors (other than 1), so we stop here.
3. Multiply all the divisors and remaining numbers
Now, we multiply all the numbers we divided by (2, 3, and 3) and the final row (3 and 5) to get the LCM:
2 × 3 × 3 × 3 × 5 = 270
So, the LCM of 54 and 90 is 270!
The division method is great for numbers with many common factors, as it makes the process easier to follow.
However, if two numbers don’t have many common factors, like 68 and 98, the division method won’t simplify them much. In these cases, prime factorization can be a more effective approach.
And there you have it. That’s all three methods.
Each one has its own strengths, so the best method to use depends on the numbers you're working with.
At Mathnasium, we break down challenging concepts into manageable steps and provide personalized support to help students truly understand math.
We’ve now got three methods in our arsenal. All we need to do is practice, practice, practice! Let’s go through a few more examples together.
Let’s find the LCM of 10 and 15 using the listing multiples method.
List the multiples of 10: 10, 20, 30, 40, 50, …
List the multiples of 15: 15, 30, 45, 60, …
Find the smallest multiple they both have: 30
So, using the listing multiples method, we found that the LCM of 10 and 15 is 30.
Let’s find the LCM of 32 and 40 using the prime factorization method.
1. Break each number into prime factors:
32 = 2 × 2 × 2 × 2 × 2 or (2⁵)
40 = 2 × 2 × 2 × 5 or (2³ × 5)
2. Take the highest power of each prime:
The prime factors we find in 32 and 40 are 2 and 5.
Highest power of 2 = 2⁵ (from 32)
Highest power of 5 = 5 (from 40)
3. Multiply the selected prime factors:
2⁵ × 5 = 32 × 5 = 160
So, using the prime factorization method, we found that the LCM of 32 and 40 is 160.
Let’s find the LCM of 90 and 126 using the division method (ladder method).
1. Write the numbers side by side:
We start by writing 90 and 126 side by side.
2. Keep dividing by the smallest common factor until none are left:
The smallest number that divides both 90 and 126 is 2, so we divide:
Now, both 45 and 63 are divisible by 3, so we divide again:
Both 15 and 21 are still divisible by 3, so we divide again:
Now, 5 and 7 have no common factors other than 1, so we stop here.
3. Multiply all divisors and the remaining numbers
Now, we multiply all the numbers we divided by (2, 3, and 3) and the final row (5 and 7) to get the LCM:
2 × 3 × 3 × 5 × 7 = 630
So, using the division method, we found that the LCM of 54 and 90 is 270!
Both least common multiple (LCM) and greatest common factor (GCF) are terms used in multiplication and division. They help us compare numbers, but they’re used for different types of problems.
LCM (least common multiple) helps us find a shared multiple—a number that two or more numbers can multiply into.
GCF (greatest common factor) helps us find a shared factor—a number that two or more numbers can be divided by.
We use LCM for:
Finding common denominators when adding or subtracting fractions.
Solving word problems about repeating events (like figuring out when two schedules line up).
For example, to add \(\frac{1}{8} + \frac{1}{12}\), we need a common denominator.
To find their common denominator, we’re really looking for their LCM—or the smallest number both 8 and 12 can divide into.
We list their multiples:
Multiples of 8: 8, 16, 24, 32, 40…
The LCM of 8 and 12 is 24.
To make 24 the denominator of \(\frac{1}{8}\), we multiply the denominator by 3, so we also multiply the numerator by 3:
\(\frac{1 \times 3}{8 \times 3} = \frac{3}{24}\)
To make 24 the denominator of \(\frac{1}{12}\), we multiply the denominator by 2, so we also multiply the numerator by 2:
\(\frac{1 \times 2}{12 \times 2} = \frac{2}{24}\)
Now, we just add the numerators:
\(\frac{3}{24} + \frac{2}{24} = \frac{5}{24}\)
We use GCF for:
Simplifying fractions by dividing both the numerator and denominator.
Dividing things into equal groups (like evenly splitting items into sets).
For example, to simplify \(\frac{16}{24}\), we divide both the numerator and denominator by the GCF of 16 and 24, which is 8.
\(\frac{16}{8} \div \frac{24}{8} = \frac{2}{3}\)
Using the GCF, we found the simplest form of the fraction, \(\frac{2}{3}\).
Check out our video guide to the least common multiple and greatest common factor:
Great job so far! It’s time to put your knowledge of LCM to the test. Try solving these tasks on your own.
When you’re done, scroll to the bottom of the guide to check your answers.
Find the LCM of 14 and 22 using the listing multiples method.
Find the LCM of 36 and 45 using the prime factorization method.
Find the LCM of 56 and 72 using the division (ladder) method.
Learning about LCM doesn’t come without dilemmas. We’ve put together a list of questions students often ask when learning this concept.
Yes! This happens when one number is a multiple of the other.
For example, if we find the LCM of 6 and 18, we list their multiples:
Multiples of 6: 6, 12, 18, 24, 30...
Multiples of 18: 18, 36, 54...
Since 18 is already a multiple of 6, the LCM of 6 and 18 is 18.
Their LCM is just their product because they have no shared prime factors.
For example, let’s find the LCM of 7 and 9:
Prime factors of 7: 7 (it’s already prime)
Prime factors of 9: 3 × 3
Since 7 and 9 have no shared prime factors, their LCM is just 7 × 9 = 63.
This happens anytime two numbers are relatively prime, meaning their GCF is 1.
No, never! The LCM is always equal to or greater than the largest number—it can’t be smaller.
For example, the LCM of 8 and 12 is 24, which is larger than both 8 and 12:
Multiples of 8: 8, 16, 24, 32...
Multiples of 12: 12, 24, 36...
Since LCM finds a shared multiple, it will always be at least as large as the biggest number.
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Here are the solutions to the LCM practice exercises—see how you did!
1. LCM(14, 22) – Listing Multiples Method
List the multiples of 14: 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154...
List the multiples of 22: 22, 44, 66, 88, 110, 132, 154...
The smallest multiple they share is 154.
LCM(14, 22) = 154
2. LCM(36, 45) – Prime Factorization Method
Step 1: Find the prime factorization
36 = 2² × 3²
45 = 3² × 5
Step 2: Take the highest power of each factor
LCM = 2² × 3² × 5
LCM = 4 × 9 × 5 = 180
LCM(36, 45) = 180
3. LCM(56, 72) – Division (Ladder) Method
Step 1: Write both numbers side by side and divide by common prime factors:
Step 2: Multiply all divisors and leftover numbers
LCM = 2 × 2 × 2 × 7 × 9 = 504
LCM(56, 72) = 504
If you got them all right, amazing work! If not, go back and try again—practice makes perfect!
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