Basic Mathematics for College Students

96 Chapter 1 Whole Numbers

In Example 7, we found that the GCF of 18 and 45 is 9. Note that 9 is the

greatest number that divides 18 and 45.

In general, the greatest common factor of two (or more) numbers is the largest

number that divides them exactly. For this reason, the greatest common factor is also

known as the greatest common divisor (GCD)and we can write GCD (18, 45)9.

45

9

 5

18

9

 2

4 Find the GCF using prime factorization.

We can find the GCF of two (or more) numbers by listing the factors of each

number. However, this method can be lengthy. Another way to find the GCF uses

the prime factorization of each number.

Finding the GCF Using Prime Factorization

To find the greatest common factor of two (or more) whole numbers:

1. Prime factor each number.


2. Identify the common prime factors.


3. The GCF is a product of all the common prime factors found in Step 2. If
   there are no common prime factors, the GCF is 1.

EXAMPLE (^8) Find the GCF of 48 and 72.
StrategyWe will begin by finding the prime factorizations of 48 and 72.
WHYThen we can identify any prime factors that they have in common.
Self Check 8
Find the GCF of 36 and 60.
Now TryProblem 57
Solution
Step 1Prime factor 48 and 72.

72  2  2  2  3  3

48  2  2  2  2  3

72

98

3324

22

48

4 12

2 2 4 3

2 2

Solution

To find the factors of 18, we find all the pairs of whole numbers whose product is

18. There are three possibilities:

To find the factors of 45, we find all the pairs of whole numbers whose product is

45. There are three possibilities:

The factors of 18 and 45 are listed below. Their common factors are circled.

Factors of 18: 1 , 2, 3 , 6, 9 , 18

Factors of 45: 1 , 3 , 5 , 9 , 15 , 45

The common factors of 18 and 45 are and 9. Since 9 is their largest common

factor,

GCF (18, 45) 9 Read as “The greatest common factor of 18 and 45 is 9.”

1, 3,

1  45  45 3  15  45 5  9  45

1  18  18 2  9  18 3  6  18