<< Chapter < Page Chapter >> Page >
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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

1

Got questions? Get instant answers now!

Exercises for review

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

Got questions? Get instant answers now!

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

53

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( [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"} {}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
how did I we'll learn this
Noor Reply
f(x)= 2|x+5| find f(-6)
Prince Reply
f(n)= 2n + 1
Samantha Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
7hours 36 min - 4hours 50 min
Tanis Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of mathematics' conversation and receive update notifications?

Ask