MATH 205 Course Notes- Section 4.3: Least Common Multiple
Ex. 1: George vacuums the rugs every 18 days, mows the grass every 12 days, and pays the bills
every 15 days. Today he did all three. How long will it be before he has another day that he does
all three?
Ex. 2: Ann and Bob are cycling on a track. Ann completes one lap every 12 seconds, and Bob
completes one lap every 15 seconds. When will Ann lap Bob, assuming that they started together?
What are we really asking for mathematically in the above? We are really talking about the least
common multiple of the numbers, or the LCM. This means that smallest number that is a multiple
of all the numbers given, or we could say the smallest number that is divisible by all the numbers
given.
Let’s investigate some special circumstances of finding the LCM.
Ex. 3: Find LCM (12, 20).
Ex. 4: Find LCM (10, 20).
Ex. 5: Find LCM (10, 21).
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Are there any observations you can make about the LCM of two numbers?
Observations: Most of the time, the LCM is greater than both of the numbers, in special cases it
is equal to one of the numbers. When does this occur?
LCM is always less than or equal to the product of the two numbers. In special cases, LCM is
equal to the product of the two numbers. When does this occur?
This second case occurs when the numbers are relatively prime, which means that the numbers
have no factors in common, other than 1. How can we determine this easily? We look at the prime
factorizations. If there are no primes in common, there will be no factors in common (other than
1), and so the numbers will be relatively prime.
Ex. 6: Find two numbers that are relatively prime.
Ex. 7: If two numbers a and b are relatively prime, find LCM (a, b).
Ex. 8: If b is divisible by a, find LCM (a, b).
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Strategies for Finding LCM
We have several strategies for finding LCM, similar to those for finding GCF.
1. Write out the multiples of each number and find the first one they have in common.
2. Find the prime factorizations of each number, and include all primes in the LCM, with the max
power that is in either of the originals. 3. Use Venn Diagrams with the prime factorizations. The
LCM is the union of the two (or three) circles, so it includes all possible primes to their highest
powers.
Ex. 9: Use strategy 1 to find LCM (30, 75).
Ex. 10: Use strategy 2 to find LCM (25, 40).
Ex. 11: Use strategy 3 to find LCM (462, 630), and LCM (40, 84, 90).
Ex. 12: If the GCF of two numbers is 18, and the LCM is 630, find the two numbers.
Ex. 13: The GCF of 66 and x is 11, and the LCM of 66 and x is 858. Find x.
What is the relationship between LCM and GCF? Answer:
LCM (a, b) =
3
a·b
GCF (a, b)