Basic Mathematics for College Students

1.8 The Least Common Multiple and the Greatest Common Factor 97

EXAMPLE (^9) Find the GCF of 8 and 15.
StrategyWe will begin by finding the prime factorizations of 8 and 15.
WHYThen we can identify any prime factors that they have in common.
Solution
The prime factorizations of 8 and 15 are shown below.
Since there are no common factors, the GCF of 8 and 15 is 1. Thus,
GCF (8, 15) 1 Read as “The greatest common factor of 8 and 15 is 1.”

15  3  5

8  2  2  2

Self Check 9

Find the GCF of 8 and 25.

Now TryProblem 61

EXAMPLE (^10) Find the GCF of and 140.
StrategyWe will begin by finding the prime factorizations of and 140.
WHYThen we can identify any prime factors that they have in common.
Solution
The prime factorizations of 20, 60, and 140 are shown below.
The circling above shows that and 140 have three common factors: two
common factors of 2 and one common factor of 5. The GCF is the product of the
circled prime factors.
Read as “The greatest common factor
of and 140 is 20.”
Note that 20 is the greatest number that divides and 140 exactly.
140
20

 7

60

20

 3

20

20

 1

20, 60,

20, 60,

GCF (20, 60, 140) 2  2  5  20

20, 60,

140  2  2  5  7

60  2  2  3  5

20  2  2  5

20, 60,

20, 60,

Self Check 10

Find the GCF of

and 75.

Now TryProblem 67

45, 60,

EXAMPLE (^11) Bouquets A florist wants to use 12 white tulips, 30 pink
tulips, and 42 purple tulips to make as many identical arrangements as possible.
Each bouquet is to have the same number of each color tulip.
a.What is the greatest number of arrangements that she can make?
b.How many of each type of tulip can she use in each bouquet?
StrategyWe will find the GCF of and 42.
WHYSince an equal number of tulips of each color will be used to create the
identical arrangements, division is indicated. The greatest common factor of three
numbers is the largest number that divides them exactly.

12, 30,

Step 2The circling on the previous page shows that 48 and 72 have four common
prime factors: Three common factors of 2 and one common factor of 3.

Step 3The GCF is the product of the circled prime factors.

GCF (48, 72) 2  2  2  3  24

Self Check 11

SCHOOL SUPPLIES A bookstore

manager wants to use some

leftover items (36 markers,

54 pencils, and 108 pens) to make

identical gift packs to donate to

an elementary school.

a.What is the greatest number

of gift packs that can be

made? (continued)