3.4: The Greatest Common Factor
- be able to find the greatest common factor of two or more whole numbers
The Greatest Common Factor (GCF)
Using the method we studied in [link] , we could obtain the prime factorizations of 30 and 42.
\[ \begin{align*} 30 &= 2 \cdot 3 \cdot 5 \\[4pt] 42 &= 2 \cdot 3 \cdot 7 \end{align*}\]
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.
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.
- 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.
Find the GCF of the following numbers.
12 and 18
Solution
- \(\begin{array} {l} {12 = 2 \cdot 6 = 2 \cdot 2 \cdot 3 = 2^2 \cdot 3} \\ {18 = 2 \cdot 9 = 2 \cdot 3 \cdot 3 = 2 \cdot 3^2} \end{array}\)
- 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.
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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.
18, 60, and 72
Solution
- \(\begin{array} {l} {18 = 2 \cdot 9 = 2 \cdot 3 \cdot 3 = 2 \cdot 3^2} \\ {60 = 2 \cdot 30 = 2 \cdot 2 \cdot 15 = 2 \cdot 2 \cdot 3 \cdot 5 = 2^2 \cdot 3 \cdot 5} \\ {72 = 2 \cdot 36 = 2 \cdot 2 \cdot 18 = 2 \cdot 2 \cdot 2 \cdot 9 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 = 2^3 \cdot 3^2} \end{array}\)
- The common bases are 2 and 3.
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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
Solution
- \(\begin{array} {rcl} {700 \ = \ 2 \cdot 350 \ = \ 2 \cdot 2 \cdot 175} & = & {2 \cdot 2 \cdot 5 \cdot 35} \\ {} & = & {2 \cdot 2 \cdot 5 \cdot 5 \cdot 7} \\ {} & = & {2^2 \cdot 5^2 \cdot 7} \\ {1,880 \ = \ 2 \cdot 940 \ = \ 2 \cdot 2 \cdot 470} & = & {2 \cdot 2 \cdot 2 \cdot 235} \\ {} & = & {2 \cdot 2 \cdot 2 \cdot 5 \cdot 47} \\ {} & = & {2^3 \cdot 5 \cdot 47} \\ {6,160 \ = \ 2 \cdot 3,080 \ = \ 2 \cdot 2 \cdot 1,540} & = & {2 \cdot 2 \cdot 2 \cdot 770} \\ {} & = & {2 \cdot 2 \cdot 2 \cdot 2 \cdot 385} \\ {} & = & {2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 77} \\ {} & = & {2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 7 \cdot 11} \\ {} & = & {2^4 \cdot 5 \cdot 7 \cdot 11} \end{array}\)
- The common bases are 2 and 5
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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 \(2^2 \cdot 5 = 4 \cdot 5 = 20\)
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.
24 and 36
- Answer
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12
Practice Set A
48 and 72
- Answer
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24
Practice Set A
50 and 140
- Answer
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10
Practice Set A
21 and 225
- Answer
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3
Practice Set A
450, 600, and 540
- Answer
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30
Exercises
For the following problems, find the greatest common factor (GCF) of the numbers.
Exercise \(\PageIndex{1}\)
6 and 8
- Answer
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2
Exercise \(\PageIndex{2}\)
5 and 10
Exercise \(\PageIndex{3}\)
8 and 12
- Answer
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4
Exercise \(\PageIndex{4}\)
9 and 12
Exercise \(\PageIndex{5}\)
20 and 24
- Answer
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4
Exercise \(\PageIndex{6}\)
35 and 175
Exercise \(\PageIndex{7}\)
25 and 45
- Answer
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5
Exercise \(\PageIndex{8}\)
45 and 189
Exercise \(\PageIndex{9}\)
66 and 165
- Answer
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33
Exercise \(\PageIndex{10}\)
264 and 132
Exercise \(\PageIndex{11}\)
99 and 135
- Answer
-
9
Exercise \(\PageIndex{12}\)
65 and 15
Exercise \(\PageIndex{13}\)
33 and 77
- Answer
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11
Exercise \(\PageIndex{14}\)
245 and 80
Exercise \(\PageIndex{15}\)
351 and 165
- Answer
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3
Exercise \(\PageIndex{16}\)
60, 140, and 100
Exercise \(\PageIndex{17}\)
147, 343, and 231
- Answer
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7
Exercise \(\PageIndex{18}\)
24, 30, and 45
Exercise \(\PageIndex{19}\)
175, 225, and 400
- Answer
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25
Exercise \(\PageIndex{20}\)
210, 630, and 182
Exercise \(\PageIndex{21}\)
14, 44, and 616
- Answer
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2
Exercise \(\PageIndex{22}\)
1,617, 735, and 429
Exercise \(\PageIndex{23}\)
1,573, 4,862, and 3,553
- Answer
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11
Exercise \(\PageIndex{24}\)
3,672, 68, and 920
Exercise \(\PageIndex{25}\)
7, 2,401, 343, 16, and 807
- Answer
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1
Exercise \(\PageIndex{26}\)
500, 77, and 39
Exercise \(\PageIndex{27}\)
441, 275, and 221
- Answer
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1
Exercises for Review
Exercise \(\PageIndex{28}\)
Find the product. \(2,753 \times 4,006\)
Exercise \(\PageIndex{29}\)
Find the quotient. \(954 \div 18\)
- Answer
-
53
Exercise \(\PageIndex{30}\)
Specify which of the digits 2, 3, or 4 divide into 9,462.
Exercise \(\PageIndex{31}\)
Write \(8 \times 8 \times 8 \times 8 \times 8 \times 8\) using exponents.
- Answer
-
\(8^6 = 262,144\)
Exercise \(\PageIndex{32}\)
Find the prime factorization of 378.