1.5 Greatest Common Factor And Least Common Multiple Worksheet With Answers - College Of The Sequoias Page 6
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11d.
Begin by listing the multiples of 4, 6, and 8:
multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, …
multiples of 6: 6, 12, 18, 24, 30, 36, 42, …
multiples of 8: 8, 16, 24, 32, 40, 48, 54, …
Since 24 is the smallest number common to all three lists, the LCM if 4, 6,
and 8 is 24.
We will now develop an approach to find the least common multiple of two numbers using prime
factorizations. Recall earlier we found the GCF of 84 and 120. The prime factorizations of those
two numbers were:
(
)
(
)
84 = 4 • 21 = 2 • 2
= 2 • 2 • 3 • 7
• 3 • 7
(
)
(
)
(
)
(
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120 = 10 • 12 = 2 • 5
= 2 • 5
= 2 • 2 • 2 • 3 • 5
• 4 • 3
• 2 • 2 • 3
Any multiple of 84 must contain the primes of 84, thus any multiple of 84 must contain the
primes 2 • 2 • 3 • 7 . Similarly, any multiple of 120 must contain the primes of 120, thus it must
contain the primes 2 • 2 • 2 • 3 • 5 . Since the LCM must be common to both list, it must contain
both sets of primes. What is the smallest number possible? It must have three 2’s (from the
second list), a 3, a 5, and a 7, so it is 2 • 2 • 2 • 3 • 5 • 7 = 840 . Thus the LCM of 84 and 120 is
840. The procedure is the same using three numbers. Suppose we want to find the LCM of 48,
60, and 150. Using 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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)
(
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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
For the LCM to be a multiple of 48, its prime factorization must contain four 2’s and one 3. To
be a multiple of 60, it must also contain a 5 (the four 2’s and one 3 take care of the rest of 60). To
be a multiple of 150, it needs another 5, so it is 2 • 2 • 2 • 2 • 3 • 5 • 5 = 1, 200 . Thus the LCM of
48, 60, and 150 is 1,200.
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