3.5: The Least Common Multiple
- Page ID
- 48847
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- be able to find the least common multiple of two or more whole numbers
Multiples
When a whole number is multiplied by other whole numbers, with the exception of zero, the resulting products are called multiples of the given whole number. Note that any whole number is a multiple of itself.
| Multiples of 2 | Multiples of 3 | Multiples 8 | Multiples of 10 |
|---|---|---|---|
| \(2 \times 1 = 2\) | \(3 \times 1 = 3\) | \(8 \times 1 = 8\) | \(10 \times 1 = 10\) |
| \(2 \times 2 = 4\) | \(3 \times 2 = 6\) | \(8 \times 2 = 16\) | \(10 \times 2 = 20\) |
| \(2 \times 3 = 6\) | \(3 \times 3 = 9\) | \(8 \times 3 = 24\) | \(10 \times 3 = 30\) |
| \(2 \times 4 = 8\) | \(3 \times 4 = 12\) | \(8 \times 4 = 32\) | \(10 \times 4 = 40\) |
| \(2 \times 5 = 10\) | \(3 \times 5 = 15\) | \(8 \times 5 = 40\) | \(10 \times 5 = 50\) |
| ... | ... | ... | ... |
Practice Set A
Find the first five multiples of the following numbers.
4
- Answer
-
4, 8, 12, 16, 20
Practice Set A
5
- Answer
-
5, 10, 15, 20, 25
Practice Set A
6
- Answer
-
6, 12, 18, 24, 30
Practice Set A
7
- Answer
-
7, 14, 21, 28, 35
Practice Set A
9
- Answer
-
9, 18, 27, 36, 45
Common Multiples
There will be times when we are given two or more whole numbers and we will need to know if there are any multiples that are common to each of them. If there are, we will need to know what they are. For example, some of the multiples that are common to 2 and 3 are 6, 12, and 18.
We can visualize common multiples using the number line.
Notice that the common multiples can be divided by both whole numbers.
Practice Set B
Find the first five common multiples of the following numbers.
2 and 4
- Answer
-
4, 8, 12, 16, 20
Practice Set B
3 and 4
- Answer
-
12, 24, 36, 48, 60
Practice Set B
2 and 5
- Answer
-
10, 20, 30, 40, 50
Practice Set B
3 and 6
- Answer
-
6, 12, 18, 24, 30
Practice Set B
4 and 5
- Answer
-
20, 40, 60, 80, 100
The Least Common Multiple (LCM)
Notice that in our number line visualization of common multiples (above), the first common multiple is also the smallest, or least common multiple, abbreviated by LCM.
The least common multiple (LCM) of two or more whole numbers is the smallest whole number that each of the given numbers will divide into without a remainder.
The least common multiple will be extremely useful in working with fractions .
Finding the Least Common Multiple
Finding the LCM
To find the LCM of two or more numbers:
- Write the prime factorization of each number, using exponents on repeated factors.
- Write each base that appears in each of the prime factorizations.
- To each base, attach the largest exponent that appears on it in the prime factorizations.
- The LCM is the product of the numbers found in step 3.
There are some major differences between using the processes for obtaining the GCF and the LCM that we must note carefully:
The Difference Between the Processes for Obtaining the GCF and the LCM
- Notice the difference between step 2 for the LCM and step 2 for the GCF. For the GCF, we use only the bases that are commonin the prime factorizations, whereas for the LCM, we use each base that appears in the prime factorizations.
- Notice the difference between step 3 for the LCM and step 3 for the GCF. For the GCF, we attach the smallest exponents to the common bases, whereas for the LCM, we attach the largest exponents to the bases.
Find the LCM of the following numbers.
9 and 12
Solution
- \(\begin{array} {l} {9 = 3 \cdot 3 = 3^2} \\ {12 = 2 \cdot 6 = 2 \cdot 2 \cdot 3 = 2^2 \cdot 3} \end{array}\)
- The bases that appear in the prime factorizations are 2 and 3.
- The largest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 2 and 2:
\(2^2\) from 12.
\(3^2\) from 9. - The LCM is the product of these numbers.
LCM = \(2^2 \cdot 3^2 = 4 \cdot 9 = 36\)
Thus, 36 is the smallest number that both 9 and 12 divide into without remainders.
90 and 630
Solution
- \(\begin{array} {ccll} {90} & = & {2 \cdot 45 = 2 \cdot 3 \cdot 15 = 2 \cdot 3 \cdot 3 \cdot 5 = 2 \cdot 3^2 \cdot 5} & {} \\ {630} & = & {2 \cdot 315 = 2 \cdot 3 \cdot 105 = 2 \cdot 3 \cdot 3 \cdot 35} & {= 2 \cdot 3 \cdot 3 \cdot 5 \cdot 7} \\ {} & \ & {} & {= 2 \cdot 3^2 \cdot 5 \cdot 7} \end{array}\)
- The bases that appear in the prime factorizations are 2, 3, 5, and 7.
- The largest exponents that appear on 2, 3, 5, and 7 are, respectively, 1, 2, 1, and 1:
\(2^1\) from either 90 or 630.
\(3^2\) from either 90 or 630.
\(5^1\) from either 90 or 630.
\(7^1\) from 630. - The LCM is the product of these numbers.
LCM = \(2 \cdot 3^2 \cdot 5 \cdot 7 = 2 \cdot 9 \cdot 5 \cdot 7 = 630\)
Thus, 630 is the smallest number that both 90 and 630 divide into with no remainders.
33, 110, and 484
Solution
- \(\begin{array} {rcl} {33} & = & {3 \cdot 11} \\ {110} & = & {2 \cdot 55 = 2 \cdot 5 \cdot 11} \\ {484} & = & {2 \cdot 242 = 2 \cdot 2 \cdot 121 = 2 \cdot 2 \cdot 11 \cdot 11 = 2^2 \cdot 11^2} \end{array}\)
- The bases that appear in the prime factorizations are 2, 3, 5, and 11.
- The largest exponents that appear on 2, 3, 5, and 11 are, respectively, 2, 1, 1, and 2:
\(2^2\) from 484.
\(3^1\) from 33.
\(5^1\) from 110.
\(11^2\) from 484. - The LCM is the product of these numbers.
\(\begin{array} {rcl} {\text{LCM}} & = & {2^2 \cdot 3 \cdot 5 \cdot 11^2} \\ {} & = & {4 \cdot 3 \cdot 5 \cdot 121} \\ {} & = & {7260} \end{array}\)
Thus, 7260 is the smallest number that 33, 110, and 484 divide into without remainders.
Practice Set C
Find the LCM of the following numbers.
20 and 54
- Answer
-
540
Practice Set C
14 and 28
- Answer
-
28
Practice Set C
6 and 63
- Answer
-
126
Practice Set C
28, 40, and 98
- Answer
-
1,960
Practice Set C
16, 27, 125, and 363
- Answer
-
6,534,000
Exercises
For the following problems, find the least common multiple of the numbers.
Exercise \(\PageIndex{1}\)
8 and 12
- Answer
-
24
Exercise \(\PageIndex{2}\)
6 and 15
Exercise \(\PageIndex{3}\)
8 and 10
- Answer
-
40
Exercise \(\PageIndex{4}\)
10 and 14
Exercise \(\PageIndex{5}\)
4 and 6
- Answer
-
12
Exercise \(\PageIndex{6}\)
6 and 12
Exercise \(\PageIndex{7}\)
9 and 18
- Answer
-
18
Exercise \(\PageIndex{8}\)
6 and 8
Exercise \(\PageIndex{9}\)
5 and 6
- Answer
-
30
Exercise \(\PageIndex{10}\)
7 and 8
Exercise \(\PageIndex{11}\)
3 and 4
- Answer
-
12
Exercise \(\PageIndex{12}\)
2 and 9
Exercise \(\PageIndex{13}\)
7 and 9
- Answer
-
63
Exercise \(\PageIndex{14}\)
28 and 36
Exercise \(\PageIndex{15}\)
24 and 36
- Answer
-
72
Exercise \(\PageIndex{16}\)
28 and 42
Exercise \(\PageIndex{17}\)
240 and 360
- Answer
-
720
Exercise \(\PageIndex{18}\)
162 and 270
Exercise \(\PageIndex{19}\)
20 and 24
- Answer
-
120
Exercise \(\PageIndex{20}\)
25 and 30
Exercise \(\PageIndex{21}\)
24 and 54
- Answer
-
216
Exercise \(\PageIndex{22}\)
16 and 24
Exercise \(\PageIndex{23}\)
36 and 48
- Answer
-
144
Exercise \(\PageIndex{24}\)
24 and 40
Exercise \(\PageIndex{25}\)
15 and 21
- Answer
-
105
Exercise \(\PageIndex{26}\)
50 and 140
Exercise \(\PageIndex{27}\)
7, 11, and 33
- Answer
-
231
Exercise \(\PageIndex{28}\)
8, 10, and 15
Exercise \(\PageIndex{29}\)
18, 21, and 42
- Answer
-
126
Exercise \(\PageIndex{30}\)
4, 5, and 21
Exercise \(\PageIndex{31}\)
45, 63, and 98
- Answer
-
4,410
Exercise \(\PageIndex{32}\)
15, 25, and 40
Exercise \(\PageIndex{33}\)
12, 16, and 20
- Answer
-
240
Exercise \(\PageIndex{34}\)
84 and 96
Exercise \(\PageIndex{35}\)
48 and 54
- Answer
-
432
Exercise \(\PageIndex{36}\)
12, 16, and 24
Exercise \(\PageIndex{37}\)
12, 16, 24, and 36
- Answer
-
144
Exercise \(\PageIndex{38}\)
6, 9, 12, and 18
Exercise \(\PageIndex{39}\)
8, 14, 28, and 32
- Answer
-
224
Exercise \(\PageIndex{40}\)
18, 80, 108, and 490
Exercise \(\PageIndex{41}\)
22, 27, 130, and 225
- Answer
-
193,050
Exercise \(\PageIndex{42}\)
38, 92, 115, and 189
Exercise \(\PageIndex{43}\)
8 and 8
- Answer
-
8
Exercise \(\PageIndex{44}\)
12, 12, and 12
Exercise \(\PageIndex{45}\)
3, 9, 12, and 3
- Answer
-
36
Exercises for Review
Exercise \(\PageIndex{46}\)
Round 434,892 to the nearest ten thousand.
Exercise \(\PageIndex{47}\)
How much bigger is 14,061 than 7,509?
- Answer
-
6,552
Exercise \(\PageIndex{48}\)
Find the quotient. \(22,428 \div 14\).
Exercise \(\PageIndex{49}\)
Expand \(84^7\). Do not find the value.
- Answer
-
\(84 \cdot 84 \cdot 84\)
Exercise \(\PageIndex{50}\)
Find the greatest common factor of 48 and 72.