*Contributor: Ashley Nail Murphy. Lesson ID: 13409*

What is an LCM? Well, it's the least common multiple of two given numbers! In this lesson, learn what this important type of number is, how to find it, and how to use it to solve real-world problems!

categories

subject

Math

learning style

Auditory, Visual

personality style

Beaver, Golden Retriever

Grade Level

Middle School (6-8)

Lesson Type

Quick Query

Alice and her best friend Micah both work jobs that keep them busy. They are planning to hang out soon when their days off line up. Alice has an off day every 8 days, and Micah is off every 12 days.

- How many days before Alice and Micah end up with the same day off?
- How can Alice and Micah make plans?
- What math skill can we use to help them?

We can use Least Common Multiple to solve this problem!

- What do you know about multiples?

(If you need to review this concept, visit the lesson found under **Additional Resources** in the right-hand sidebar.)

Multiples are the product of multiplying a given number by an integer. Multiples are also what you get when skip counting.

For example, look at the multiples of 2:

** 2** (2x1), **4** (2x2), **6** (2x3), **8** (2x4), **10** (2x5), **12** (2x6) ...

This list can go on forever.

Look at the multiples of 3:

** 3** (3x1), **6** (3x2), **9** (3x3), **12** (3x4), **15** (3x5), **18** (3x6) …

Correct! 6 is the Least Common Multiple of 2 and 3. The answer can be written like this:

LCM (2,3) = 6

Least Common Multiples are easy to find when the given numbers are smaller, like 2 and 3. As the numbers get larger, there are two strategies to use to find the LCM.

Let's try both strategies with Alice and Micah's story problem:

LCM (8,12) = ?

**Prime Factorization**

The prime factorization method is best when the given numbers are larger.

For this method, we break the given number down by its factors until the factors left are only the number and 1.

Look at 8 first.

8 can be broken into its factors of 2 and 4.

2 cannot be broken down anymore because 2's only factors are 1 and itself. 4 can be broken down further into 2 and 2.

Take the dead ends of the factorization and write a multiplication equation:

To check you used prime factorization correctly, make sure 2 x 2 x 2 does in fact equal 8.

Now, repeat the prime factorization for 12.

12 can be split into the factors of 2 and 6.

6 can be further broken into 2 and 3.

Now, to find the LCM of 8 and 12, we need to combine the multiplication equations to include the factors of both given numbers.

The new equation includes 2 x 2 x 2 to represent 8 and 2 x 2 x 3 to represent 12.

Since both equations include two 2s, those are overlapped and not included twice.

Last, multiply the factors in the combined equation:

LCM (8,12) = 2 x 2 x 2 x 3

LCM (8, 12) = 24

**List Multiples**

The second strategy to find the Least Common Multiple of two given numbers is to simply list the multiples.

Let's look at 8 and 12 again.

First, list the multiples for 8. Multiple 8 by each integer starting with 1:

** 8** (8x1), **16** (8x2), **24** (8x3), **32** (8x4), **40** (8x5), **48** (8x6), **56** (8x7) ...

Then, make a list of multiples for 12:

**12** (12x1), **24** (12x2), **36** (12x3), **48** (12x4), **60** (12x5), **72** (12x6), **85** (12x7), **96** (12x8) ...

Next, find any common multiples:

8, 16, **24**, 32, 40, 48, 56, 64, 72, 80, 88, **96** ...

12, **24**, 36, 48, 60, 72, 85, **96**, 108 ...

Last, identify the least common multiple:

LCM (8,12) = 24

We solved for the same answer with both methods!

Review these two methods of finding the LCM with *Least common multiple exercise | Factors and multiples | Pre-Algebra | Khan Academy*:

Consider the following questions:

- Which strategy do you think you like better?
- Are there certain situations where one method is better than the other?
- How can finding the LCM of a given number be a useful math skill?

To practice using both strategies, click NEXT to visit the *Got It?* section.