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Using the method we studied in [link] , we could obtain the prime factorizations of 30 and 42.
$\text{30}=2\cdot 3\cdot 5$
$\text{42}=2\cdot 3\cdot 7$
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.
Find the GCF of the following numbers.
12 and 18
$\begin{array}{c}\text{12}=2\cdot 6=2\cdot 2\cdot 3={2}^{2}\cdot 3\\ \text{18}=2\cdot 9=2\cdot 3\cdot 3=2\cdot {3}^{2}\end{array}$
$2\cdot 3=6$
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
$\begin{array}{c}\text{18}=2\cdot 9=2\cdot 3\cdot 3=2\cdot {3}^{2}\hfill \\ \text{60}=2\cdot \text{30}=2\cdot 2\cdot \mathrm{15}=2\cdot 2\cdot 3\cdot 5={2}^{2}\cdot 3\cdot 5\hfill \\ \text{72}=2\cdot \text{36}=2\cdot 2\cdot \text{18}=2\cdot 2\cdot 2\cdot 9=2\cdot 2\cdot 2\cdot 3\cdot 3={2}^{3}\cdot {3}^{2}\hfill \end{array}$
The 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.
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 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.
Find the GCF of the following numbers.
For the following problems, find the greatest common factor (GCF) of the numbers.
35 and 175
45 and 189
264 and 132
65 and 15
245 and 80
60, 140, and 100
24, 30, and 45
210, 630, and 182
1,617, 735, and 429
3,672, 68, and 920
500, 77, and 39
( [link] ) Find the product. $\mathrm{2,}\text{753}\times \mathrm{4,}\text{006}$ .
( [link] ) Find the quotient. $\text{954}\xf7\text{18}$ .
53
( [link] ) Specify which of the digits 2, 3, or 4 divide into 9,462.
( [link] ) Write $8\times 8\times 8\times 8\times 8\times 8$ using exponents.
${8}^{6}=\text{262},\text{144}$
( [link] ) Find the prime factorization of 378.
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