1.5 Greatest Common Factor And Least Common Multiple Worksheet With Answers - College Of The Sequoias Page 3
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11Recall that the GCF is a factor of each number. But remember from the previous section that all
factors of a number must be composed of its primes. Thus the factors of 84 are all comprised of
some combination of 2 (can be used twice), 3, and 7. Similarly, the factors of 120 are all
comprised of some combination of 2 (can be used three times), 3, and 5. Since the GCF is
common to both lists, how large can it be? It could have 2 (used twice) and 3, or 2 • 2 • 3 = 12 .
Thus the GCF of 84 and 120 is 12. The procedure is the same using three numbers. Suppose we
want to find the GCF of 48, 60, and 150. First find the prime factorizations of the three numbers:
(
)
(
)
(
)
(
)
48 = 4 • 12 = 2 • 2
= 2 • 2
= 2 • 2 • 2 • 2 • 3
• 3 • 4
• 3 • 2 • 2
(
)
(
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60 = 10 • 6 = 2 • 5
= 2 • 2 • 3 • 5
• 2 • 3
(
)
(
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150 = 10 • 15 = 2 • 5
= 2 • 3 • 5 • 5
• 3 • 5
Looking at the three lists, we are looking for the largest set of primes common to all three lists.
This is 2 • 3 = 6 , so the GCF of 48, 60, and 150 is 6.
Example 2
Find the greatest common factor of each set of numbers by using primes.
a.
48, 80
b.
60, 150
c.
168, 210
d.
45, 60, 180
Solution
a.
Begin by finding the prime factorization of each number:
(
)
(
)
(
)
(
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48 = 8 • 6 = 2 • 4
= 2 • 2 • 2
= 2 • 2 • 2 • 2 • 3
• 2 • 3
• 2 • 3
(
)
(
)
(
)
(
)
80 = 10 • 8 = 2 • 5
= 2 • 5
= 2 • 2 • 2 • 2 • 5
• 2 • 4
• 2 • 2 • 2
The largest set of primes common to both lists is 2 • 2 • 2 • 2 = 16 , so the
GCF of 48 and 80 is 16.
b.
Begin by finding the prime factorization of each number:
(
)
(
)
60 = 10 • 6 = 2 • 5
= 2 • 2 • 3 • 5
• 2 • 3
(
)
(
)
150 = 10 • 15 = 2 • 5
= 2 • 3 • 5 • 5
• 3 • 5
The largest set of primes common to both lists is 2 • 3 • 5 = 30 , so the
GCF of 60 and 150 is 30.
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