5.5: The Least Common Multiple
- Page ID
- 51011
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.