



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:
 Write the prime factorization of each number, using exponents on repeated factors.
 Write each base that is common to each of the numbers.
 To each base listed in step 2, attach the
smallest exponent that appears on it in either of the prime factorizations.
 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

$\begin{array}{c}\text{12}=2\cdot 6=2\cdot 2\cdot 3={2}^{2}\cdot 3\\ \text{18}=2\cdot 9=2\cdot 3\cdot 3=2\cdot {3}^{2}\end{array}$
 The common bases are 2 and 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.
 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.
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18, 60, and 72

$\begin{array}{c}\text{18}=2\cdot 9=2\cdot 3\cdot 3=2\cdot {3}^{2}\hfill \\ \text{60}=2\cdot \text{30}=2\cdot 2\cdot \mathrm{15}=2\cdot 2\cdot 3\cdot 5={2}^{2}\cdot 3\cdot 5\hfill \\ \text{72}=2\cdot \text{36}=2\cdot 2\cdot \text{18}=2\cdot 2\cdot 2\cdot 9=2\cdot 2\cdot 2\cdot 3\cdot 3={2}^{3}\cdot {3}^{2}\hfill \end{array}$
 The common bases are 2 and 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

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.
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700, 1,880, and 6,160
$$

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

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.

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.
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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.
Exercises for review
Questions & Answers
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
sure. what is your question?
ninjadapaul
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X6)^2
so it's 20 divided by X6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x6
ninjadapaul
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ninjadapaul
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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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.
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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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