Greatest Common Factor and Least Common
Multiple
MATH 100 Survey of Mathematical Ideas
J. Robert Buchanan
Department of Mathematics
Fall 2014
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Greatest Common Factor
Definition
The greatest common factor (GCF) (sometimes called the
greatest common divisor (GCD)) of a group of natural
numbers is the largest natural number that is a factor of all the
numbers in the group.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Greatest Common Factor
Definition
The greatest common factor (GCF) (sometimes called the
greatest common divisor (GCD)) of a group of natural
numbers is the largest natural number that is a factor of all the
numbers in the group.
Prime Factors Method: to find the GCF
1
Write the prime factorization of each number.
2
Choose all the primes common to all factorizations, with
each prime raised to the least exponent that it has in any
factorization.
3
Form the product of all the numbers in Step 2; this product
is the GCF.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
180 and 300
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
180 and 300
180 = (22 )(32 )(5)
300 = (22 )(3)(52 )
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
180 and 300
180 = (22 )(32 )(5)
300 = (22 )(3)(52 )
gcd(180, 300) = (22 )(3)(5) = 60
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
180 and 300
180 = (22 )(32 )(5)
300 = (22 )(3)(52 )
gcd(180, 300) = (22 )(3)(5) = 60
252 and 308 and 504
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
180 and 300
180 = (22 )(32 )(5)
300 = (22 )(3)(52 )
gcd(180, 300) = (22 )(3)(5) = 60
252 and 308 and 504
252 = (22 )(32 )(7)
308 = (22 )(7)(11)
504 = (23 )(32 )(7)
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
180 and 300
180 = (22 )(32 )(5)
300 = (22 )(3)(52 )
gcd(180, 300) = (22 )(3)(5) = 60
252 and 308 and 504
252 = (22 )(32 )(7)
308 = (22 )(7)(11)
504 = (23 )(32 )(7)
gcd(252, 308, 504) = (22 )(7) = 28
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Alternative Methods
Dividing by Prime Factors Method: to find the GCF of a list of
numbers,
1
Write the numbers in a row.
2
Divide each of the numbers by a common prime factor (try
2, then 3, and so on).
3
Divide the quotients by a common prime factor. Continue
until no prime will divide into all the quotients.
4
The product of the primes in Steps 2 and 3 is the GCF.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
130 and 455
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
130 and 455
130 455
÷ 5 26
91
÷ 13
2
7
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
130 and 455
130 455
÷ 5 26
91
÷ 13
2
7
gcd(130, 455) = (5)(13) = 65
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
130 and 455
130 455
÷ 5 26
91
÷ 13
2
7
gcd(130, 455) = (5)(13) = 65
432 and 450 and 1500
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
130 and 455
130 455
÷ 5 26
91
÷ 13
2
7
gcd(130, 455) = (5)(13) = 65
432 and 450 and 1500
÷2
÷3
432
216
72
J. Robert Buchanan
450
225
75
1500
750
250
Greatest Common Factor and Least Common Multiple
Examples
Find the GCF of the following lists of numbers.
130 and 455
130 455
÷ 5 26
91
÷ 13
2
7
gcd(130, 455) = (5)(13) = 65
432 and 450 and 1500
÷2
÷3
432
216
72
450
225
75
1500
750
250
gcd(432, 450, 1500) = (2)(3) = 6
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Example
Use your i>clicker2 to enter the greatest common factor of the
following lists of numbers.
166 and 415
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Example
Use your i>clicker2 to enter the greatest common factor of the
following lists of numbers.
166 and 415
384 and 444
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Example
Use your i>clicker2 to enter the greatest common factor of the
following lists of numbers.
166 and 415
384 and 444
450, 630 and 1155
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Example
Use your i>clicker2 to enter the greatest common factor of the
following lists of numbers.
166 and 415
384 and 444
450, 630 and 1155
138, 184 and 437
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Euclidean Algorithm
Euclidean Algorithm: to find the GCF of a pair of unequal
numbers, divide the larger number by the smaller number. Note
the remainder and divide the previous divisor by this remainder.
Continue the process until a remainder of 0 is obtained. The
GCF is the last positive remainder obtained in the process.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Example
Find the GCF of 25 and 70.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Example
Find the GCF of 25 and 70.
a
25
J. Robert Buchanan
b
70
r
20
Greatest Common Factor and Least Common Multiple
Example
Find the GCF of 25 and 70.
a
25
20
J. Robert Buchanan
b
70
25
r
20
5
Greatest Common Factor and Least Common Multiple
Example
Find the GCF of 25 and 70.
a
25
20
5
J. Robert Buchanan
b
70
25
20
r
20
5
0
Greatest Common Factor and Least Common Multiple
Example
Find the GCF of 25 and 70.
a
25
20
5
b
70
25
20
r
20
5
0
gcd(25, 70) = 5
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Application
Question: A carpenter has some pieces of two-by-four lumber.
Some are 60 inches long and some are 72 inches long. All of
them must be cut into shorter pieces. If all the cut pieces must
be the same length, what is the longest such piece so that no
lumber is left over?
Use your i>clicker2 to submit your answer.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Least Common Multiple
Definition
The least common multiple (LCM) of a group of natural
numbers is the smallest natural number that is a multiple of all
the numbers in the group.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Least Common Multiple
Definition
The least common multiple (LCM) of a group of natural
numbers is the smallest natural number that is a multiple of all
the numbers in the group.
Prime Factors Method: to find the LCM
1
Write the prime factorization of each number.
2
Choose all the primes belonging to any factorization, with
each prime raised to the largest exponent that it has in any
factorization.
3
Form the product of all the numbers in Step 2; this product
is the LCM.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
12 and 32
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
12 and 32
12 = (22 )(3)
32 = 25
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
12 and 32
12 = (22 )(3)
32 = 25
lcm(12, 32) = (25 )(3) = 96
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
12 and 32
12 = (22 )(3)
32 = 25
lcm(12, 32) = (25 )(3) = 96
24 and 36 and 48
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
12 and 32
12 = (22 )(3)
32 = 25
lcm(12, 32) = (25 )(3) = 96
24 and 36 and 48
24 = (23 )(3)
36 = (22 )(32 )
48 = (24 )(3)
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
12 and 32
12 = (22 )(3)
32 = 25
lcm(12, 32) = (25 )(3) = 96
24 and 36 and 48
24 = (23 )(3)
36 = (22 )(32 )
48 = (24 )(3)
lcm(24, 36, 48) = (24 )(32 ) = 144
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Alternative Methods
Dividing by Prime Factors Method: to find the LCM of a list of
numbers,
1
Write the numbers in a row.
2
Divide each of the numbers by a common prime factor (try
2, then 3, and so on).
3
Divide the quotients by a common prime factor. When no
prime will divide all quotients, but a prime will divide some
of them, divide where possible and bring any non-divisible
quotients down. Continue until no prime will divide any two
quotients.
4
The product of all the prime divisors from Steps 2 and 3 as
well as all remaining quotients is the LCM.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
35 and 56
48 and 54 and 60
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
35 and 56
÷7
35
5
56
8
48 and 54 and 60
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
35 and 56
÷7
35
5
56
8
lcm(35, 56) = (7)(5)(8) = 280
48 and 54 and 60
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
35 and 56
÷7
35
5
56
8
lcm(35, 56) = (7)(5)(8) = 280
48 and 54 and 60
÷2
÷3
÷2
J. Robert Buchanan
48
24
8
4
54
27
9
60
30
10
5
Greatest Common Factor and Least Common Multiple
Examples
Find the LCM of the following lists of numbers.
35 and 56
÷7
35
5
56
8
lcm(35, 56) = (7)(5)(8) = 280
48 and 54 and 60
÷2
÷3
÷2
48
24
8
4
54
27
9
60
30
10
5
lcm(48, 54, 60) = (2)(2)(3)(4)(5)(9) = 2160
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Example
Find the least common multiple of the following lists of
numbers. Submit your answers using your i>clicker2.
12 and 34
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Example
Find the least common multiple of the following lists of
numbers. Submit your answers using your i>clicker2.
12 and 34
35 and 100
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Example
Find the least common multiple of the following lists of
numbers. Submit your answers using your i>clicker2.
12 and 34
35 and 100
36, 48 and 60
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Example
Find the least common multiple of the following lists of
numbers. Submit your answers using your i>clicker2.
12 and 34
35 and 100
36, 48 and 60
12, 28 and 150
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Formula
The LCM of two natural numbers m and n is
LCM(m, n) =
J. Robert Buchanan
m·n
.
GCF(m, n)
Greatest Common Factor and Least Common Multiple
Formula
The LCM of two natural numbers m and n is
LCM(m, n) =
m·n
.
GCF(m, n)
Find the LCM of 130 and 455.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Formula
The LCM of two natural numbers m and n is
LCM(m, n) =
m·n
.
GCF(m, n)
Find the LCM of 130 and 455.
LCM(130, 455) =
(130)(455)
59150
=
= 910
gcd(130, 455)
65
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Application
Kathryn Campbell and Tami Dreyfus are in a bicycle race on a
circular track. If they start at the same place and travel in the
same direction and Kathryn completes a revolution every 40
seconds while Tami completes a revolution every 45 seconds,
how long will it take them before they reach the starting point
again simultaneously?
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Application
Kathryn Campbell and Tami Dreyfus are in a bicycle race on a
circular track. If they start at the same place and travel in the
same direction and Kathryn completes a revolution every 40
seconds while Tami completes a revolution every 45 seconds,
how long will it take them before they reach the starting point
again simultaneously?
LCM(40, 45) =
1800
(40)(45)
=
= 360
gcd(40, 45)
5
J. Robert Buchanan
seconds
Greatest Common Factor and Least Common Multiple
Application
Chuck and Buck work as security guards at a university. Chuck
has every sixth night off, and Buck every tenth night off. If both
are off on July 1, what is the next night that they will both be off
together?
S
1
8
15
22
29
M
2
9
16
23
30
T
3
10
17
24
31
JUL
W
T
4
5
11 12
18 19
25 26
F
6
13
20
27
S
7
14
21
28
S
M
T
5
12
19
26
6
13
20
27
7
14
21
28
AUG
W
T
1
2
8
9
15 16
22 23
29 30
F
3
10
17
24
31
S
4
11
18
25
S
M
T
SEP
W
T
F
2
9
16
23
30
3
10
17
24
4
11
18
25
5
12
19
26
6
13
20
27
7
14
21
28
S
1
8
15
22
29
Use your i>clicker2 to submit the date in the format MMDD.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple
Relatively Prime Numbers
If a and b are integers and gcd(a, b) = 1 then a and b are
said to be relatively prime.
Integers a and b do not have to be prime to be relatively
prime.
If a and b are relatively prime then lcm(a, b) = a b.
J. Robert Buchanan
Greatest Common Factor and Least Common Multiple