3.4 The greatest common factor

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses the greatest common factor. By the end of the module students should be able to find the greatest common factor of two or more whole numbers.

Section overview

• The Greatest Common Factor (GCF)
• A Method for Determining the Greatest Common Factor

The greatest common factor (gcf)

Using the method we studied in [link] , we could obtain the prime factoriza­tions of 30 and 42.

$\text{30}=2\cdot 3\cdot 5$

$\text{42}=2\cdot 3\cdot 7$

Common factor

We notice that 2 appears as a factor in both numbers, that is, 2 is a common factor of 30 and 42. We also notice that 3 appears as a factor in both numbers. Three is also a common factor of 30 and 42.

Greatest common factor (gcf)

When considering two or more numbers, it is often useful to know if there is a largest common factor of the numbers, and if so, what that number is. The largest common factor of two or more whole numbers is called the greatest common factor , and is abbreviated by GCF . The greatest common factor of a collection of whole numbers is useful in working with fractions (which we will do in [link] ).

A method for determining the greatest common factor

A straightforward method for determining the GCF of two or more whole numbers makes use of both the prime factorization of the numbers and exponents.

Finding the gcf

To find the greatest common factor (GCF) of two or more whole numbers:
1. Write the prime factorization of each number, using exponents on repeated factors.
2. Write each base that is common to each of the numbers.
3. To each base listed in step 2, attach the smallest exponent that appears on it in either of the prime factorizations.
4. The GCF is the product of the numbers found in step 3.

Sample set a

Find the GCF of the following numbers.

12 and 18

1. $\begin{array}{c}\text{12}=2\cdot 6=2\cdot 2\cdot 3={2}^{2}\cdot 3\\ \text{18}=2\cdot 9=2\cdot 3\cdot 3=2\cdot {3}^{2}\end{array}$

2. The common bases are 2 and 3.
3. The smallest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 1 and 1 ( ${2}^{1}$ and ${3}^{1}$ ), or 2 and 3.
4. The GCF is the product of these numbers.

$2\cdot 3=6$

The GCF of 30 and 42 is 6 because 6 is the largest number that divides both 30 and 42 without a remainder.

18, 60, and 72

1. $\begin{array}{c}\text{18}=2\cdot 9=2\cdot 3\cdot 3=2\cdot {3}^{2}\hfill \\ \text{60}=2\cdot \text{30}=2\cdot 2\cdot \mathrm{15}=2\cdot 2\cdot 3\cdot 5={2}^{2}\cdot 3\cdot 5\hfill \\ \text{72}=2\cdot \text{36}=2\cdot 2\cdot \text{18}=2\cdot 2\cdot 2\cdot 9=2\cdot 2\cdot 2\cdot 3\cdot 3={2}^{3}\cdot {3}^{2}\hfill \end{array}$

2. The common bases are 2 and 3.
3. The smallest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 1 and 1:

${2}^{1}$ from 18

${3}^{1}$ from 60

4. The GCF is the product of these numbers.

GCF is $2\cdot 3=6$

Thus, 6 is the largest number that divides 18, 60, and 72 without a remainder.

700, 1,880, and 6,160 

1. $\begin{array}{ccccccc}\mathrm{700}& =& 2\cdot \mathrm{350}& =& 2\cdot 2\cdot \mathrm{175}& =& 2\cdot 2\cdot 5\cdot \mathrm{35}\hfill \\ & & & & & =& 2\cdot 2\cdot 5\cdot 5\cdot 7\hfill \\ & & & & & =& {2}^{2}\cdot {5}^{2}\cdot 7\hfill \\ \mathrm{1,880}& =& 2\cdot \mathrm{940}& =& 2\cdot 2\cdot \mathrm{470}& =& 2\cdot 2\cdot 2\cdot \mathrm{235}\hfill \\ & & & & & =& 2\cdot 2\cdot 2\cdot 5\cdot \mathrm{47}\hfill \\ & & & & & =& {2}^{3}\cdot 5\cdot \mathrm{47}\hfill \\ \mathrm{6,160}& =& 2\cdot \mathrm{3,080}& =& 2\cdot 2\cdot \mathrm{1,540}& =& 2\cdot 2\cdot 2\cdot \mathrm{770}\hfill \\ & & & & & =& 2\cdot 2\cdot 2\cdot 2\cdot \mathrm{385}\hfill \\ & & & & & =& 2\cdot 2\cdot 2\cdot 2\cdot 5\cdot \mathrm{77}\hfill \\ & & & & & =& 2\cdot 2\cdot 2\cdot 2\cdot 5\cdot 7\cdot \mathrm{11}\hfill \\ & & & & & =& {2}^{4}\cdot 5\cdot 7\cdot \mathrm{11}\hfill \end{array}$

2. The common bases are 2 and 5
3. The smallest exponents appearing on 2 and 5 in the prime factorizations are, respectively, 2 and 1.

${2}^{2}$ from 700.

${5}^{1}$ from either 1,880 or 6,160.

4. The GCF is the product of these numbers.

GCF is ${2}^{2}\cdot 5=4\cdot 5=\text{20}$

Thus, 20 is the largest number that divides 700, 1,880, and 6,160 without a remainder.

Practice set a

Find the GCF of the following numbers.

24 and 36

12

48 and 72

24

50 and 140

10

21 and 225

3

450, 600, and 540

30

Exercises

For the following problems, find the greatest common factor (GCF) of the numbers.

6 and 8

2

5 and 10

8 and 12

4

9 and 12

20 and 24

4

35 and 175

25 and 45

5

45 and 189

66 and 165

33

264 and 132

99 and 135

9

65 and 15

33 and 77

11

245 and 80

351 and 165

3

60, 140, and 100

147, 343, and 231

7

24, 30, and 45

175, 225, and 400

25

210, 630, and 182

14, 44, and 616

2

1,617, 735, and 429

1,573, 4,862, and 3,553

11

3,672, 68, and 920

7, 2,401, 343, 16, and 807

1

500, 77, and 39

441, 275, and 221

1

Exercises for review

( [link] ) Find the product. $2,\text{753}×4,\text{006}$ .

( [link] ) Find the quotient. $\text{954}÷\text{18}$ .

53

( [link] ) Specify which of the digits 2, 3, or 4 divide into 9,462.

( [link] ) Write $8×8×8×8×8×8$ using exponents.

${8}^{6}=\text{262},\text{144}$

( [link] ) Find the prime factorization of 378.

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