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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.

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

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

    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.

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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 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:

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

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

  4. The GCF is the product of these numbers.

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

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

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

  1. 700 = 2 350 = 2 2 175 = 2 2 5 35 = 2 2 5 5 7 = 2 2 5 2 7 1,880 = 2 940 = 2 2 470 = 2 2 2 235 = 2 2 2 5 47 = 2 3 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 = 2 4 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.

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

    5 1 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 2 2 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.

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Practice set a

Find the GCF of the following numbers.

Exercises

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

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

11

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

1

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Exercises for review

( [link] ) Find the product. 2, 753 × 4, 006 size 12{2,"753" times 4,"006"} {} .

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( [link] ) Find the quotient. 954 ÷ 18 size 12{"954" div "18"} {} .

53

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

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( [link] ) Write 8 × 8 × 8 × 8 × 8 × 8 size 12{8´8´8´8´8´8} {} using exponents.

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

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

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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