5.5: The Least Common Multiple

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Page ID: 51011

Amy Lagusker
College of the Canyons

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Definition: Least Common Multiple (LCM)

The LCM is the smallest number that is a multiple of all numbers, excluding zero.

To find the LCM, we work backwards, following the three steps below.

Example \(\PageIndex{1}\)

Find the LCM of 54 and 30

Solution

Step 1: List the Multiples

54 → 54, 108, 162, 216, 270, 324,…

30 → 30, 60, 90, 120, 150, 180, 210, 240, 270, 300,…

Step 2: Circle the Common multiple(s)

Step 3: Choose the Least common multiple = 270

Partner Activity 1

A recipe for peanut butter cookies will make 15 cookies. A recipe for chocolate cookies will make two dozen cookies. If you want to have the same number of each type of cookie, what is the least number of each that you will need to make using complete recipes?

Example \(\PageIndex{2}\)

Add (and Subtract) Fractions Using the LCM and GCF

Solution

\[\dfrac{11}{12}+\dfrac{7}{20}=\dfrac{11 \times 5}{12 \times 5}+\dfrac{7 \times 3}{20 \times 3}=\dfrac{55}{60}+\dfrac{21}{60}=\dfrac{55+21}{60}=\dfrac{76}{60}=\dfrac{76 \div 4}{60 \div 4}=\dfrac{19}{15} \nonumber \]

Partner Activity 2

Add or subtract the following fractions:

\(\dfrac{5}{6}-\dfrac{3}{4}\)
\(\dfrac{2}{3}-\dfrac{1}{2}\)
\(6 \dfrac{4}{5}+7 \dfrac{1}{5}\)

Practice Problems

Find the Least Common Multiple (LCM).

40 and 90
75 and 25
168 and 85
90, 120, and 150
135, 225, and 405

Add or Subtract fractions.

\(\dfrac{7}{12}+\dfrac{8}{5}\)
\(\dfrac{3}{16}-\dfrac{2}{9}\)
\(\dfrac{34}{26}+\dfrac{10}{13}\)

Extension: Methods of Teaching Mathematics

Part 1

Using the standard and topic you choose from earlier this semester, write a full 45-minute lesson plan. See Canvas for more detailed instructions.

Part 2

Make sure you are working on Khan Academy throughout the semester.