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Least Common Multiple
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When we learned adding and subtracting fractions, we learned that the number on the bottom, the denominator, has to be the same. In order to get a common denominator, we first need to find the Least Common Multiple (LCM). The LCM of two numbers is the smallest number that both numbers divide evenly. The video below will explain two methods of finding LCMs:

Least Common Multiple (LCM)

Video Source (08:44 mins) | Transcript

The two ways to find the LCM:


  1. Skip Counting
    • List the multiples of each of the numbers given and find the smallest number in both lists.
  2. Prime Factorization
    • Find all the prime factors of each number given.
    • Create a new number that contains all the prime factors of each number. Remember to include multiples if there are multiples of the same factor in either prime factorization. Example: \(9 = 3 \times 3\) and \(15 = 3 \times 5\), since 9 has two 3s and 15 has only one 3 in its factorization, the combined list will need two 3s. The LCM of 9 and 15 is \( 3 \times 3 \times 5 = 45\)

Additional Resources

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), we will 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, we can write the prime factorization of 6 as \(2 \times 3\).

    We have the following:
    \(5 = 5\)
    \(6 = 2 \times 3\)

    Our 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), we will 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, we 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 rightSince 2 and 3 are both prime, we can write the prime factorization of 12 as \(2 \times 2 \times 3\).

    We have the following:
    \(4 = 2 \times 2\)
    \(12 = 2 \times 2 \times 3\)

    Our 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, we 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


    (Video Source | Transcript)

    Detail: To find the least common multiple (LCM), we will 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, we 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, we can write the prime factorization of 10 as \(2 \times 5\).

    We have the following:
    \(6 = 2 \times 3\)
    \(10 = 2 \times 5\)

    Our 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, we 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.

    \(LCM = 2 \times 3 \times 5 = 30\)
    )
  4. 4 and 14 (
    Solution
    x
    Solution: 28
    Details: To find the least common multiple (LCM), we will 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, we 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, we can write the prime factorization of 14 as \(2 \times 7\).

    We have the following:
    \(4 = 2 \times 2\)
    \(14 = 2 \times 7\)

    Our LCM will need to have the prime factors of both 4 and 14. Since the prime factorization of 4 is \(2 \times 2\), when we include the prime factors of 14 we already have at least one 2, so we 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


    (Video Source | Transcript)

    Detail: To find the least common multiple (LCM), we will 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, we can write the prime factorization of 9 as \(3 \times 3\).

    We have the following:
    \(7 = 7\)
    \(9 = 3 \times 3\)

    Our LCM will need to have the prime factors of both 7 and 9, so we 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.

    \(LCM = 7 × 3 × 3 = 63\)
    )
  6. 7 and 5 (
    Solution
    x
    Solution: 35
    Details: To find the least common multiple (LCM), we will 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.

    We have the following:
    \(5 = 5\)
    \(7 = 7\)

    Our LCM will need to have the prime factors of both 7 and 5, so we 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.

    \(LCM = 7 \times 5 = 35\)
    )

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