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Least Common Multiple
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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)

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 \(2\times7 \times 3 =42\).

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: \(2 \times3\)
  • Prime factors of 8: \(2\times 4 = 2 \times2 \times 2\)

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

Find the least common multiple for the following pairs:
  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.

    Image with a 6 at the top and two lines pointing down and outward to a 2 on the bottom left and a 3 on the bottom right.
    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.

    The image is an equation that begins with LCM equals. The middle of the equation is 5 times 2 times 3. Above the 5, there is a bracket that points to the number 5. Below the 2 times 3, there is a bracket that points to the number 6. After the numbers in the middle, there is an equal sign with the number 30 beside it.

    \(LCM = 2 \times 3 \times 5 = 30\)
    )
  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.

    Image with 4 on top and two lines pointing down and outward to a 2 on the bottom left and a 2 on the bottom right.
    Since 2 is prime, you can write the prime factorization of 4 as \(2 \times 2\).

    Image with 12 on top and two lines pointing down and outward to a 3 on the bottom left and a 4 on the bottom right. Under the 4 are two more lines pointing down and outward to a 2 on the bottom left and a 2 on the bottom right
    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.

    This image is an equation. The first part of the equation is LCM equals. After the equal sign comes the middle of the equation, the middle of the equation reads 2 times 2 times 3. A bracket rests above the 2 times 2 and it has the number 4 above it. Another bracket resides underneath 2 times 2 times 3, and it has the number 12 below it. After the middle of the equation, the third part of the equation begins with an equal sign followed by the number 12.

    \(LCM = 2 \times 2 \times 3 = 12\)
    )
  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.

    Image with a 6 at the top and two lines pointing down and outward to a 2 on the bottom left and a 3 on the bottom right.
    Since 2 and 3 are both prime, you can write the prime factorization of 6 as \(2 \times 3\).

    Image with a 10 at the top and two lines pointing down and outward to a 2 on the bottom left and a 5 on the bottom right.
    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.

    The image is an equation written as such: LCM equals 3 times 2 times 5, equals 30. Above the 3 times 2, there is a bracket encompassing 3 and 2, and pointing to the number 6. Below the 2 times 5, there is a bracket that points to a 10.

    \(LCM = 2 \times 3 \times 5 = 30\)
     | 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.

    Image with 4 on top and two lines pointing down and outward to a 2 on the bottom left and a 2 on the bottom right.
    Since 2 is prime, you can write the prime factorization of 4 as \(2 \times 2\).

    Image with 14 on top and two lines pointing down and outward to a 2 on the bottom left and a 7 on the bottom right.
    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.

    The image is an equation written as such: LCM equals 2 times 2 times 7, equals 28. Above the 2 times 2, there is a bracket encompassing 2 and 2, and pointing to the number 4. Below the 2 times 7, there is a bracket that points to a 14.

    \(LCM = 2 \times 2 \times 7 = 28\)
    )
  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.

    Image with 9 on top and two lines pointing down and outward to a 3 on the bottom left and a 3 on the bottom right.
    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 \(3 \times 3\).

    The image is an equation written as such: LCM equals 7 times 3 times 3, equals 63. Above the 7, there is a bracket encompassing 7 and pointing to a number 7. Below the 3 times 3, there is a bracket encompassing the 3 and 3, that points to a number 9.

    \(LCM = 7 × 3 × 3 = 63\)
     | 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.

    The image is an equation written as such: LCM equals 7 times 5 equals 35. Above the 7, there is a bracket pointing to a purple 7. Below the 5, there is a bracket that points to a green 5.

    \(LCM = 7 \times 5 = 35\)
    )
Addition and Subtraction of Fractions with Common Denominators
Finding Common Denominators