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How to Find the Least Common Multiple of Expressions

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  • 0:02 Least Common Multiple (LCM)
  • 0:49 Example 1
  • 3:25 Example 2
  • 6:22 Example 3
  • 10:23 Lesson Summary
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Lesson Transcript
Instructor: Maria Blojay

Maria has taught College Algebra and has a master's degree in Education Administration.

Would you like to learn an easy method for finding the least common multiple of expressions? In this lesson, we will look at sample problems with prime factorization (a product of prime numbers) to explain the steps.

Least Common Multiple (LCM)

The LCM of expressions is the smallest expression that can be divided by each of the other expressions. In other words, the LCM is the smallest multiple in common with each of the expressions. You've probably worked with this concept before when finding least common denominators. Now we are applying this concept to expressions.

There are three steps:

  1. Write the factors for each expression in prime factorization and count how often each factor occurs.
  2. Identify each factor's most frequent occurrence.
  3. Highlight all most frequently occurring factors, and then find the product of the highlighted factors. The product is the LCM.

Ready to learn how these steps work? Here are three examples:

Example 1

Given these three expressions, find the LCM:

  1. 4x2y2
  2. 6xy
  3. 10xy2

First Step:

Write the factors for each expression in prime factorization and count how often each factor occurs.

Expression 1: 4x2y2 = (2)(2)(x)(x)(y)(y)
The prime factorization of 4 = (2)(2).

  • 2 occurs two times.
  • Variable x occurs two times.
  • Variable y occurs two times.

Expression 2: 6xy = (2)(3)(x)(y)
The prime factorization of 6 = (2)(3)

  • 2 occurs one time.
  • 3 occurs one time.
  • Variable x occurs one time.
  • Variable y occurs one time.

Expression 3: 10xy2 = (2)(5)(x)(y)(y)
The prime factorization of 10 = (2)(5)

  • 2 occurs one time.
  • 5 occurs one time.
  • Variable x occurs one time.
  • Variable y occurs two times.

Second Step:

Now, identify each factor's most frequent occurrence.

Out of the expressions given in Example 1:

  • The 2 occurs the most in Expression 1.
  • The 3 occurs in Expression 2 one time and zero times in the other expressions; so, the 3 occurs the most in Expression 2.
  • The 5 occurs in Expression 3 one time and zero times in the other expressions; so, the 5 occurs the most in Expression 3.
  • The variable x occurs the most in Expression 1.
  • The variable y occurs the most in Expressions 1 and 3, causing a tie, so we can just pick either Expression 1 or 3. Let's choose Expression 3.

Third Step:

Highlight in bold print all most frequently occurring factors.

Expression 1: 4x2y2 = (2)(2)(x)(x)(y)(y)

Expression 2: 6xy = (2)(3)(x)(y)

Expression 3: 10xy2 = (2)(5)(x)(y)(y)

Then write down the highlighted factors and find the product, which will be the LCM:

(2)(2)(3)(5)(x)(x)(y)(y) = 60x2y2 is the LCM.

Example 2

Given these three expressions, find the LCM:

  1. 4a2c2
  2. 15a5c7
  3. 16ac

First Step:

Our variables this time are a and c. Write the factors for each expression in prime factorization and count how often each factor occurs.

Expression 1: 4a2c2 = (2)(2)(a)(a)(c)(c)
The prime factorization of 4 = (2)(2).

  • 2 occurs two times.
  • Variable a occurs two times.
  • Variable c occurs two times.

Expression 2: 15a5c7 = (3)(5)(a)(a)(a)(a)(a)(c)(c)(c)(c)(c)(c)(c)
The prime factorization of 15 = (3)(5)

  • 3 occurs one time.
  • 5 occurs one time.
  • Variable a occurs five times.
  • Variable c occurs seven times.

Expression 3: 16ac = (2)(2)(2)(2)(a)(c)
The prime factorization of 16 = (2)(2)(2)(2)

  • 2 occurs four times.
  • Variable a occurs one time.
  • Variable c occurs one time.

Second Step:

Now, identify each factor's most frequent occurrence.

Out of the expressions given in Example 2:

  • The 2 occurs the most in Expression 3.
  • The 3 occurs in Expression 2 one time and zero times in the other expressions; so, the 3 occurs the most in Expression 2.
  • The 5 occurs in Expression 2 one time and zero times in the other expressions; so, the 5 occurs the most in Expression 2.
  • The variable a occurs the most in Expression 2.
  • The variable c occurs the most in Expression 2.

Third Step:

Highlight all most frequently occurring factors.

Expression 1: 4a2c2 = (2)(2)(a)(a)(c)(c)

Expression 2: 15a5c7 = (3)(5)(a)(a)(a)(a)(a)(c)(c)(c)(c)(c)(c)(c)

Expression 3: 16ac = (2)(2)(2)(2)(a)(c)

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