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The Greatest Common Factor

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: 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 (Reference), we could obtain the prime factoriza­tions of 30 and 42.

30 = 2 ⋅ 3 ⋅ 5 30 = 2 ⋅ 3 ⋅ 5 size 12{"30"=2 cdot 3 cdot 5} {}

42 = 2 ⋅ 3 ⋅ 7 42 = 2 ⋅ 3 ⋅ 7 size 12{"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 (Reference)).

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.

Example 1

12 and 18

  1. 12 = 2 ⋅ 6 = 2 ⋅ 2 ⋅ 3 = 2 2 ⋅ 3 18 = 2 ⋅ 9 = 2 ⋅ 3 ⋅ 3 = 2 ⋅ 3 2 12 = 2 ⋅ 6 = 2 ⋅ 2 ⋅ 3 = 2 2 ⋅ 3 size 12{"12"=2 cdot 6=2 cdot 2 cdot 3=2 rSup { size 8{2} } cdot 3} {} 18 = 2 ⋅ 9 = 2 ⋅ 3 ⋅ 3 = 2 ⋅ 3 2 size 12{"18"=2 cdot 9=2 cdot 3 cdot 3=2 cdot 3 rSup { size 8{2} } } {}

  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 ( 2121 size 12{2 rSup { size 8{1} } } {}and 3131 size 12{3 rSup { size 8{1} } } {}), or 2 and 3.
  4. The GCF is the product of these numbers.

    2 ⋅ 3 = 6 2 ⋅ 3 = 6 size 12{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.

Example 2

18, 60, and 72

  1. 18 = 2 ⋅ 9 = 2 ⋅ 3 ⋅ 3 = 2 ⋅ 3 2 60 = 2 ⋅ 30 = 2⋅2⋅15= 2 ⋅ 2 ⋅ 3 ⋅ 5 = 2 2 ⋅ 3 ⋅ 5 72 = 2 ⋅ 36 = 2 ⋅ 2 ⋅ 18 = 2 ⋅ 2 ⋅ 2 ⋅ 9 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 = 2 3 ⋅ 3 2 18 = 2 ⋅ 9 = 2 ⋅ 3 ⋅ 3 = 2 ⋅ 3 2 60 = 2 ⋅ 30 = 2⋅2⋅15= 2 ⋅ 2 ⋅ 3 ⋅ 5 = 2 2 ⋅ 3 ⋅ 5 72 = 2 ⋅ 36 = 2 ⋅ 2 ⋅ 18 = 2 ⋅ 2 ⋅ 2 ⋅ 9 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 = 2 3 ⋅ 3 2 alignl { stack { size 12{"18"=2 cdot 9=2 cdot 3 cdot 3=2 cdot 3 rSup { size 8{2} } } {} # "60"=2 cdot "30"=2 cdot 2 cdot 3 cdot 5=2 rSup { size 8{2} } cdot 3 cdot 5 {} # "72"=2 cdot "36"=2 cdot 2 cdot "18"=2 cdot 2 cdot 2 cdot 9=2 cdot 2 cdot 2 cdot 3 cdot 3=2 rSup { size 8{3} } cdot 3 rSup { size 8{2} } {} } } {}

  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:

    2121 size 12{2 rSup { size 8{1} } } {} from 18

    3131 size 12{3 rSup { size 8{1} } } {} from 60

  4. The GCF is the product of these numbers.

    GCF is 2⋅3=62⋅3=6 size 12{2 cdot 3=6} {}

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

Example 3

700, 1,880, and 6,160 {}

  1. 700 = 2⋅350 = 2⋅2⋅175 = 2⋅2⋅5⋅35 = 2⋅2⋅5⋅5⋅7 = 22⋅52⋅7 1,880 = 2⋅940 = 2⋅2⋅470 = 2⋅2⋅2⋅235 = 2⋅2⋅2⋅5⋅47 = 23⋅5⋅47 6,160 = 2⋅3,080 = 2⋅2⋅1,540 = 2⋅2⋅2⋅770 = 2⋅2⋅2⋅2⋅385 = 2⋅2⋅2⋅2⋅5⋅77 = 2⋅2⋅2⋅2⋅5⋅7⋅11 = 24⋅5⋅7⋅11 700 = 2⋅350 = 2⋅2⋅175 = 2⋅2⋅5⋅35 = 2⋅2⋅5⋅5⋅7 = 22⋅52⋅7 1,880 = 2⋅940 = 2⋅2⋅470 = 2⋅2⋅2⋅235 = 2⋅2⋅2⋅5⋅47 = 23⋅5⋅47 6,160 = 2⋅3,080 = 2⋅2⋅1,540 = 2⋅2⋅2⋅770 = 2⋅2⋅2⋅2⋅385 = 2⋅2⋅2⋅2⋅5⋅77 = 2⋅2⋅2⋅2⋅5⋅7⋅11 = 24⋅5⋅7⋅11

  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.

    2222 size 12{2 rSup { size 8{2} } } {} from 700.

    5151 size 12{5 rSup { size 8{1} } } {} from either 1,880 or 6,160.

  4. The GCF is the product of these numbers.

    GCF is 22⋅5=4⋅5=2022⋅5=4⋅5=20 size 12{2 rSup { size 8{2} } cdot 5=4 cdot 5="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.

Exercise 1

24 and 36

Solution

12

Exercise 2

48 and 72

Solution

24

Exercise 3

50 and 140

Solution

10

Exercise 4

21 and 225

Solution

3

Exercise 5

450, 600, and 540

Solution

30

Exercises

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

Exercise 6

6 and 8

Solution

2

Exercise 7

5 and 10

Exercise 8

8 and 12

Solution

4

Exercise 9

9 and 12

Exercise 10

20 and 24

Solution

4

Exercise 11

35 and 175

Exercise 12

25 and 45

Solution

5

Exercise 13

45 and 189

Exercise 14

66 and 165

Solution

33

Exercise 15

264 and 132

Exercise 16

99 and 135

Solution

9

Exercise 17

65 and 15

Exercise 18

33 and 77

Solution

11

Exercise 19

245 and 80

Exercise 20

351 and 165

Solution

3

Exercise 21

60, 140, and 100

Exercise 22

147, 343, and 231

Solution

7

Exercise 23

24, 30, and 45

Exercise 24

175, 225, and 400

Solution

25

Exercise 25

210, 630, and 182

Exercise 26

14, 44, and 616

Solution

2

Exercise 27

1,617, 735, and 429

Exercise 28

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

Solution

11

Exercise 29

3,672, 68, and 920

Exercise 30

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

Solution

1

Exercise 31

500, 77, and 39

Exercise 32

441, 275, and 221

Solution

1

Exercises for Review

Exercise 33

((Reference)) Find the product. 2,753×4,0062,753×4,006 size 12{2,"753" times 4,"006"} {}.

Exercise 34

((Reference)) Find the quotient. 954÷18954÷18 size 12{"954" div "18"} {}.

Solution

53

Exercise 35

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

Exercise 36

((Reference)) Write 8×8×8×8×8×88×8×8×8×8×8 size 12{8´8´8´8´8´8} {} using exponents.

Solution

86=262,14486=262,144 size 12{8 rSup { size 8{6} } ="262","144"} {}

Exercise 37

((Reference)) Find the prime factorization of 378.

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