OpenStax-CNX module: m34876
1
Exponents, Roots, Factorization of
Whole Numbers: The Least Common
Multiple
∗
Wade Ellis
Denny Burzynski
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0†
Abstract
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This
module discusses the least common multiple. By the end of the module students should be able to nd
the least common multiple of two or more whole numbers.
1 Section Overview
•
•
•
•
Multiples
Common Multiples
The Least Common Multiple (LCM)
Finding the Least Common Multiple
2 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.
∗ Version
1.2: Aug 18, 2010 8:26 pm -0500
† http://creativecommons.org/licenses/by/3.0/
http://cnx.org/content/m34876/1.2/
OpenStax-CNX module: m34876
2
2.1 Sample Set A
Multiples of 2
Multiples of 3
Multiples of 8
2×1=2
3×1=3
8×1=8
2×2=4
3×2=6
8 × 2 = 16
2×3=6
3×3=9
8 × 3 = 24
2×4=8
3 × 4 = 12
8 × 4 = 32
2 × 5 = 10
3 × 5 = 15
8 × 5 = 40
..
.
..
.
..
.
Multiples of 10
10 × 1 = 10
10 × 2 = 20
10 × 3 = 30
10 × 4 = 40
10 × 5 = 50
..
.
Table 1
2.2 Practice Set A
Find the rst ve multiples of the following numbers.
Exercise 1
(Solution on p. 8.)
Exercise 2
(Solution on p. 8.)
Exercise 3
(Solution on p. 8.)
Exercise 4
(Solution on p. 8.)
Exercise 5
(Solution on p. 8.)
4
5
6
7
9
3 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.
3.1 Sample Set B
Example 1
We can visualize common multiples using the number line.
Notice that the common multiples can be divided by both whole numbers.
http://cnx.org/content/m34876/1.2/
OpenStax-CNX module: m34876
3
3.2 Practice Set B
Find the rst ve common multiples of the following numbers.
Exercise 6
(Solution on p. 8.)
Exercise 7
(Solution on p. 8.)
Exercise 8
(Solution on p. 8.)
Exercise 9
(Solution on p. 8.)
Exercise 10
(Solution on p. 8.)
2 and 4
3 and 4
2 and 5
3 and 6
4 and 5
4 The Least Common Multiple (LCM)
Notice that in our number line visualization of common multiples (above), the rst common multiple is also
the smallest, or least common multiple, abbreviated by LCM.
Least Common Multiple
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 ().
5 Finding the Least Common Multiple
Finding the LCM
To nd the LCM of two or more numbers:
1.
2.
3.
4.
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 dierences between using the processes for obtaining the GCF and the LCM that
we must note carefully:
The Dierence Between the Processes for Obtaining the GCF and the LCM
1. Notice the dierence between step 2 for the LCM and step 2 for the GCF. For the GCF, we use only
the bases that are common in the prime factorizations, whereas for the LCM, we use each base that
appears in the prime factorizations.
2. Notice the dierence 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.
http://cnx.org/content/m34876/1.2/
OpenStax-CNX module: m34876
4
5.1 Sample Set C
Find the LCM of the following numbers.
Example 2
9 and 12
9 = 3 · 3 = 32
1.
12 = 2 · 6 = 2 · 2 · 3 = 22 · 3
2. The bases that appear in the prime factorizations are 2 and 3.
3. The largest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 2
and 2: 22 from 12.
32 from 9.
4. The LCM is the product of these numbers. LCM = 22 · 32 = 4 · 9 = 36
Thus, 36 is the smallest number that both 9 and 12 divide into without remainders.
Example 3
90 and 630
90
=
2 · 45 = 2 · 3 · 15 = 2 · 3 · 3 · 5 = 2 · 32 · 5
1. 630 =
2 · 315 = 2 · 3 · 105 = 2 · 3 · 3 · 35
=2·3·3·5·7
= 2 · 32 · 5 · 7
2. The bases that appear in the prime factorizations are 2, 3, 5, and 7.
3. The largest exponents that appear on 2, 3, 5, and 7 are, respectively, 1, 2, 1, and 1:
21 from either 90 or 630.
32 from either 90 or 630.
51 from either 90 or 630.
71 from 630.
4. The LCM is the product of these numbers. LCM = 2 · 32 · 5 · 7 = 2 · 9 · 5 · 7 = 630
Thus, 630 is the smallest number that both 90 and 630 divide into with no remainders.
Example 4
33, 110, and 484
33
3 · 11
=
1. 110 =
2 · 55 = 2 · 5 · 11
484
2 · 242 = 2 · 2 · 121 = 2 · 2 · 11 · 11 = 22 · 112 .
=
2. The bases that appear in the prime factorizations are 2, 3, 5, and 11.
3. The largest exponents that appear on 2, 3, 5, and 11 are, respectively, 2, 1, 1, and 2:
22 from 484.
31 from 33.
51 from 110
112 from 484.
LCM = 22 · 3 · 5 · 112
4. The LCM is the product of these numbers.
= 4 · 3 · 5 · 121
=
7260
Thus, 7260 is the smallest number that 33, 110, and 484 divide into without remainders.
http://cnx.org/content/m34876/1.2/
OpenStax-CNX module: m34876
5
5.2 Practice Set C
Find the LCM of the following numbers.
Exercise 11
(Solution on p. 8.)
Exercise 12
(Solution on p. 8.)
Exercise 13
(Solution on p. 8.)
Exercise 14
(Solution on p. 8.)
Exercise 15
(Solution on p. 8.)
20 and 54
14 and 28
6 and 63
28, 40, and 98
16, 27, 125, and 363
6 Exercises
For the following problems, nd the least common multiple of the numbers.
Exercise 16
8 and 12
(Solution on p. 8.)
Exercise 17
6 and 15
Exercise 18
8 and 10
(Solution on p. 8.)
Exercise 19
10 and 14
Exercise 20
4 and 6
(Solution on p. 8.)
Exercise 21
6 and 12
Exercise 22
9 and 18
(Solution on p. 8.)
Exercise 23
6 and 8
Exercise 24
5 and 6
(Solution on p. 8.)
Exercise 25
7 and 8
Exercise 26
3 and 4
(Solution on p. 8.)
Exercise 27
2 and 9
Exercise 28
7 and 9
Exercise 29
28 and 36
http://cnx.org/content/m34876/1.2/
(Solution on p. 8.)
OpenStax-CNX module: m34876
Exercise 30
24 and 36
6
(Solution on p. 8.)
Exercise 31
28 and 42
Exercise 32
240 and 360
(Solution on p. 8.)
Exercise 33
162 and 270
Exercise 34
20 and 24
(Solution on p. 8.)
Exercise 35
25 and 30
Exercise 36
24 and 54
(Solution on p. 9.)
Exercise 37
16 and 24
Exercise 38
36 and 48
(Solution on p. 9.)
Exercise 39
24 and 40
Exercise 40
15 and 21
(Solution on p. 9.)
Exercise 41
50 and 140
Exercise 42
7, 11, and 33
(Solution on p. 9.)
Exercise 43
8, 10, and 15
Exercise 44
18, 21, and 42
(Solution on p. 9.)
Exercise 45
4, 5, and 21
Exercise 46
45, 63, and 98
(Solution on p. 9.)
Exercise 47
15, 25, and 40
Exercise 48
12, 16, and 20
(Solution on p. 9.)
Exercise 49
84 and 96
Exercise 50
48 and 54
Exercise 51
12, 16, and 24
http://cnx.org/content/m34876/1.2/
(Solution on p. 9.)
OpenStax-CNX module: m34876
Exercise 52
12, 16, 24, and 36
7
(Solution on p. 9.)
Exercise 53
6, 9, 12, and 18
Exercise 54
8, 14, 28, and 32
(Solution on p. 9.)
Exercise 55
18, 80, 108, and 490
Exercise 56
22, 27, 130, and 225
(Solution on p. 9.)
Exercise 57
38, 92, 115, and 189
Exercise 58
8 and 8
(Solution on p. 9.)
Exercise 59
12, 12, and 12
Exercise 60
3, 9, 12, and 3
(Solution on p. 9.)
6.1 Exercises for Review
Exercise 61
() Round 434,892 to the nearest ten thousand.
Exercise 62
() How much bigger is 14,061 than 7,509?
(Solution on p. 9.)
Exercise 63
() Find the quotient. 22, 428 ÷ 14.
Exercise 64
() Expand 843 . Do not nd the value.
Exercise 65
() Find the greatest common factor of 48 and 72.
http://cnx.org/content/m34876/1.2/
(Solution on p. 9.)
OpenStax-CNX module: m34876
Solutions to Exercises in this Module
Solution to Exercise (p. 2)
4, 8, 12, 16, 20
Solution to Exercise (p. 2)
5, 10, 15, 20, 25
Solution to Exercise (p. 2)
6, 12, 18, 24, 30
Solution to Exercise (p. 2)
7, 14, 21, 28, 35
Solution to Exercise (p. 2)
9, 18, 27, 36, 45
Solution to Exercise (p. 3)
4, 8, 12, 16, 20
Solution to Exercise (p. 3)
12, 24, 36, 48, 60
Solution to Exercise (p. 3)
10, 20, 30, 40, 50
Solution to Exercise (p. 3)
6, 12, 18, 24, 30
Solution to Exercise (p. 3)
20, 40, 60, 80, 100
Solution to Exercise (p. 5)
540
Solution to Exercise (p. 5)
28
Solution to Exercise (p. 5)
126
Solution to Exercise (p. 5)
1,960
Solution to Exercise (p. 5)
6,534,000
Solution to Exercise (p. 5)
24
Solution to Exercise (p. 5)
40
Solution to Exercise (p. 5)
12
Solution to Exercise (p. 5)
18
Solution to Exercise (p. 5)
30
Solution to Exercise (p. 5)
12
Solution to Exercise (p. 5)
63
Solution to Exercise (p. 6)
72
Solution to Exercise (p. 6)
720
http://cnx.org/content/m34876/1.2/
8
OpenStax-CNX module: m34876
Solution to Exercise (p. 6)
120
Solution to Exercise (p. 6)
216
Solution to Exercise (p. 6)
144
Solution to Exercise (p. 6)
105
Solution to Exercise (p. 6)
231
Solution to Exercise (p. 6)
126
Solution to Exercise (p. 6)
4,410
Solution to Exercise (p. 6)
240
Solution to Exercise (p. 6)
432
Solution to Exercise (p. 7)
144
Solution to Exercise (p. 7)
224
Solution to Exercise (p. 7)
193,050
Solution to Exercise (p. 7)
8
Solution to Exercise (p. 7)
36
Solution to Exercise (p. 7)
6,552
Solution to Exercise (p. 7)
84 · 84 · 84
http://cnx.org/content/m34876/1.2/
9