Introduction

In this lesson, you will learn how to find the Least Common Multiple (LCM). In order to add and subtract fractions, the denominator has to be the same. In order to get a common denominator, you first need to find the LCM.

The LCM of two or more numbers is the smallest number that divides evenly into all the numbers.

This video illustrates the lesson material below. Watching the video is optional.

• Least Common Multiple (LCM) (08:44 mins) | Transcript

Least Common Multiple (LCM)

The LCM is used when adding and subtracting fractions with different denominators. When two numbers are given, the LCM is a multiple that both numbers share. Two methods for finding the LCM are to list the multiples and to list the prime factorization.

List the Multiples to Find the LCM

Example 1
Find the least common multiple of 3 and 5. Start by listing out the multiples of each number:

• Multiples of 3: 3, 6, 9, 12, 15, 18, …

• Multiples of 5: 5, 10, 15, 20, 25, …

The LCM is the smallest multiple that both numbers have in common. By listing the multiples of each number, you identified that both 3 and 5 have a common multiple of 15. There are no common multiples lower than 15. Therefore, 15 is the LCM of 3 and 5.

Example 2
Find the LCM of 2 and 3.

• Multiples of 2: 2, 4, 6, 8, 10, 12, 14, …

• Multiples of 3: 3, 6, 9, 12, 15, 18, …

Both numbers share the common multiple of 6 as well as 12. Since 6 is less than 12, 6 is the least common multiple. When there is more than one common multiple, always choose the smallest common multiple.

Example 3
Find the LCM of 6 and 8.

• Multiples of 6: 6, 12, 18, 24, 30, 36, …

• Multiples of 8: 8, 16, 24, 32, 40, 48, …

By listing the multiples of each number, you can identify the LCM of 6 and 8 as 24.

List the Prime Factorization to Find the LCM

Another way to find the least common multiple of two numbers is to find their prime factorization and then compare the prime factorization of both numbers.

Example 4
Find the LCM of 14 and 21 by listing all the prime factorizations:

• Prime factors of 14: $2\times 7$

• Prime factors of 21: $3\times 7$

The LCM will consist of the prime factors listed above.

1. Start with prime factors of the first number (14): $2\times 7$.
2. Include the prime factors from the second number (21) which were not included from the first number: $2\times 7\times 3$. Note: Since 7 was included as a factor of 14, you do not use it again.
3. The LCM of 14 and 21 is $2\times 7\times 3=42$.

This means 42 is the smallest common multiple of 7 and 14.
Example 5
Find the LCM of 6 and 8 by listing all the prime factorizations:

• Prime factors of 6: $2\times 3$

• Prime factors of 8: $2\times 4=2\times 2\times 2$

To find the LCM using prime factorization:

1. Start with prime factors of the first number (6): $2\times 3$.
2. Include the prime factors from the second number (8) which were not included from the first number: $2\times 3\times 2\times 2$. It has all the prime factors of 6 and also all the prime factors of 8. Note: Since you already included one 2, you just need two more 2’s from the factors of 8.
3. The LCM of 6 and 8 is $2\times 3\times 2\times 2=24$.

Finding the LCM using the prime factorization will give you the same result as listing out the multiples of each number. 24 is the least common multiple of 6 and 8, whether you find it by listing the prime factorization or listing the multiples.

Things to Remember

There are 2 different methods to find the LCM of numbers:

1. List Multiples
   
   • List the multiples of each of the numbers given and find the smallest number that's on both lists.
2. Prime Factorization
   
   • Find all the prime factors of each number given.
   • Compare the prime factors and cancel out any multiples that are shared.
   • Multiply the remaining prime factors to find the LCM.
     
     • Example: $9=3\times 3$ and $15=3\times 5$, since 9 has two 3s and 15 has only one 3 in its factorization, only one of the 3s needs to be cancelled out. The LCM of 9 and 15 is $3\times 3\times 5=45$

Practice Problems

1. 5 and 6 (
   Solution

   x

   Solution: 30
   Details:
   To find the least common multiple (LCM), start by finding the prime factors of both 5 and 6.
   
   Since 5 is prime, the prime factorization of 5 is just itself.
   
   
   Since 2 and 3 are both prime, you can write the prime factorization of 6 as $2\times 3$.
   
   You have the following:
   $5=5$
   $6=2\times 3$
   
   The LCM will need to have the prime factors of both 5 and 6, so it needs to include 2, 3, and 5.
   
   
   
   $LCM=2\times 3\times 5=30$
   )
2. 4 and 12 (
   Solution

   x

   Solution: 12
   Details:
   To find the least common multiple (LCM), start by finding the prime factors of both 4 and 12.
   
   
   Since 2 is prime, you can write the prime factorization of 4 as $2\times 2$.
   
   
   Since 2 and 3 are both prime, you can write the prime factorization of 12 as $2\times 2\times 3$.
   
   You have the following:
   $4=2\times 2$
   $12=2\times 2\times 3$
   
   The LCM will need to have the prime factors of both 4 and 12. Since the prime factorization of 4 is $2\times 2$ and the prime factorization of 12 has $2\times 2$ in it, you will only need to include $2\times 2$ once.
   
   
   
   $LCM=2\times 2\times 3=12$
   )
3. 6 and 10 (
   Video Solution

   x

   Solution: 30
   
   

   
   
   
   Details:
   To find the least common multiple (LCM), start by finding the prime factors of both 6 and 10.
   
   
   Since 2 and 3 are both prime, you can write the prime factorization of 6 as $2\times 3$.
   
   
   Since 2 and 5 are both prime, you can write the prime factorization of 10 as $2\times 5$.
   
   You have the following:
   $6=2\times 3$
   $10=2\times 5$
   
   The LCM will need to have the prime factors of both 6 and 10. Since the prime factorization of both 6 and 10 include a 2, you will only need to include 2 once.
   
   
   
   $LCM=2\times 3\times 5=30$
   | Transcript)
4. 4 and 14 (
   Solution

   x

   Solution: 28
   Details:
   To find the least common multiple (LCM), start by finding the prime factors of both 4 and 14.
   
   
   Since 2 is prime, you can write the prime factorization of 4 as $2\times 2$.
   
   
   Since 2 and 7 are both prime, you can write the prime factorization of 14 as $2\times 7$.
   
   You have the following:
   $4=2\times 2$
   $14=2\times 7$
   
   The LCM will need to have the prime factors of both 4 and 14. Since the prime factorization of 4 is $2\times 2$, when you include the prime factors of 14 you already have at least one 2, so you do not need to include another 2.
   
   
   
   $LCM=2\times 2\times 7=28$
   )
5. 7 and 9 (
   Video Solution

   x

   Solution: 63
   
   

   
   
   
   Details:
   To find the least common multiple (LCM), start by finding the prime factors of both 7 and 9.
   
   Since 7 is prime, the prime factorization of 7 is just itself.
   
   
   Since 3 is prime, you can write the prime factorization of 9 as $3\times 3$.
   
   You have the following:
   $7=7$
   $9=3\times 3$
   
   The LCM will need to have the prime factors of both 7 and 9, so you will include both 7 and $3\times 3$.
   
   
   
   $LCM=7\times 3\times 3=63$
   | Transcript)
6. 7 and 5 (
   Solution

   x

   Solution: 35
   Details:
   To find the least common multiple (LCM), start by finding the prime factors of both 7 and 5. Since 7 and 5 are both prime, the prime factorization of each is just itself.
   
   You have the following:
   $5=5$
   $7=7$
   
   The LCM will need to have the prime factors of both 7 and 5, so you will include both 7 and 5.
   
   
   
   $LCM=7\times 5=35$
   )