Learning Goal
In this lesson, you will: f Determine the least common multiple of two numbers.
Key Terms f
common multiple f least common multiple (LCM)
T ick Tock, Tick Tock!
You’ve heard clocks or watches tick, and cuckoo clocks “cuckoo.” You know that every one of those ticks represents one second, and every chime indicates the beginning of an hour. But have you ever wondered why there are 60 seconds in a minute? Or 60 minutes in an hour? Who came up with the concept of “time?”
To begin to answer these questions, you have to go all the way back to the
Sumerian civilization—starting in 5300 BC! The Sumerians used a base
60 number system. Similar to the base 10 number system, 60 and multiples of 60 were the basis of this system. This may be why we measure the repeated cycle of
60 seconds in a minute and 60 minutes in the hour. What are some things you can name that repeat in cycles?
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Students use multiples to predict the next time events will repeat in a cyclic fashion over time.
They will use common multiples to predict when two or more of these events occur at the same time. Students use two methods to determine the Least Common
Multiple (LCM). First, students will list all factors of numbers looking for the first instance of the same number. Secondly, they will use the prime factorization of each number and determine the greatest number of times each prime factor appears. Students then use these methods to answer questions about different situations.
t Ask a student to read the introduction before
Question 1 aloud. Discuss the context and complete
Questions 1 through 3 as a class.
t Ask a student to read the definition of a common multiple aloud. Have students complete
Question 4 with a partner.
Then share the responses as a class.
Many things in life repeat over and over again in a regular cycle.
●
A news radio station in Atlanta broadcasts a weather report every
12 minutes.
●
A ferry crosses Lake Champlain from Vermont to New York every
36 minutes.
●
Big Ben in London, England, chimes every 15 minutes.
Determine when each event will repeat using multiples.
1. After the radio station broadcasts a weather report at 8:00
AM
, the station will broadcast another weather report in
12 minutes, and then in
24
minutes, and then in
36
minutes . . .
2. As one ferry departs at 8:00
72
AM
minutes, and then in
, the next ferry will depart in
108
minutes . . .
36
minutes, then in
3. After Big Ben chimes at 8:00 again in
30
AM
, the clock will chime again in
minutes, and again in
45
minutes . . .
15
minutes, then
You can determine when two or more of these events will occur at the same time using common multiples . A common multiple is a number that is a multiple of two or more numbers.
4. At 8:00
AM
, the radio station broadcasted a weather report at the same time that
Big Ben chimed. a.
In how many minutes will the radio station broadcast a weather report and Big
Ben chime at the same time again?
List multiples of each and explain your reasoning.
The multiples of 12 are 12, 24, 36, 48, 60, . . .
The multiples of 15 are 15, 30, 45, 60, . . .
The radio station will broadcast a weather report and Big Ben will chime at the same time in 60 minutes.
b.
How can you continue to determine the next time both the radio station will broadcast a weather report and Big Ben will chime at the same time?
Explain your reasoning.
The list of multiples of 60 will give me the times when the radio station will broadcast a weather report and Big Ben will chime at the same time.
t How did you determine the next time the radio station broadcasts a weather report?
t How did you determine the next time the ferry departs?
t How did you determine the next time Big Ben chimes?
t t
What role do multiples play in the chiming of Big Ben?
t What role do multiples play in the timing of the weather report?
t What do multiples have to do with determining the next time Big Ben will chime at the same time the weather report broadcasts?
How much time will elapse when this event happens two times?
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› Chapter 2 Prime Factorization and the Fundamental Theorem of Arithmetic

Ask a student to read the definition of least common multiple aloud. Have students complete Question 5 with a partner. Then share the responses as a class.
t How is this question similar to Question 4?
t How is this question different than Question 4?
t What is the significance of multiples in the solution to this question?
Ask a student to read the worked example to determine the least common multiple aloud. Discuss the examples as a class.
The least common multiple , abbreviated as LCM, is the smallest multiple (other than zero) that two or more numbers have in common.
5. If a ferry departs for New York and Big Ben chimes at the same time at 12:30 PM , how can you determine when both events will occur again at the same time?
Explain your reasoning.
I can determine when these two events will take place by calculating the least common multiple.
A more efficient method to determine the least common multiple of two numbers is to use prime factors. Let’s consider using prime factorization to determine the LCM of 36 and 15.
To calculate the LCM of 36 and 15, first write the prime factors of each, and identify the unique prime factors.
36 15
4 9 3 5
The unique prime factors are 2, 3 and 5.
2 2 3 3
Next, determine the greatest power of each prime factor.
36 5 2 2 3 3 2
15 5 3 3 5
Finally, multiply those factors.
LCM 5 2 2 3 3 2 3 5
5 4 3 9 3 5
5 180
The LCM of 36 and 15 is 180.
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Have students complete
Questions 6 through 13 with a partner. Then share the responses as a class.
t Is the product of all prime factors always the same as the least common multiple of two numbers?
t When is the product of all factors the same as the least common multiple of two numbers?
t When is the product of all factors not the same as the least common multiple of two numbers?
t What is the greatest common multiple?
t What do the least common multiple and the greatest common multiple have in common?
t How do least common multiples help you determine when both school clubs meet on the same day?
6. What does LCM 5 180 mean in terms of the ferry departing for New York and Big Ben chiming?
The ferry’s departing to New York and Big Ben chiming will happen at the same time every 180 minutes.
7. Calculate the product of all the prime factors of both 15 and 36.
The product of all the prime factors is 540.
8. How does the product of all the prime factors of both 15 and 36 compare to the LCM for the ferry departing for New York and Big Ben chiming? Explain your reasoning.
The product of all the prime factors is a multiple of the LCM. The product of all the primes is three times more than the LCM.
9. What is the greatest common multiple of 15 and 36? Explain your reasoning.
I cannot determine the greatest common multiple because the multiples continue forever.
10. The Newspaper Club at Marshall Middle School meets every 6 school days. The
Math Club meets every 8 school days. Luis is a member of both clubs. He needs to make a plan when both clubs meet on the same school day. Both clubs will meet today after school.
a.
After today, when will both clubs meet on the same day again?
Explain your reasoning.
Both clubs will both meet 24 school days from today.
6 5 2 3 3
8 5 2 3
The LCM is 2 3 3 3 5 24.
b.
How often will Luis have both club meetings on the same school day?
Luis will have both club meetings on the same day every 24 school days.
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How do least common multiples help you determine when the cafeteria serves two or three menu items on the same day?
11. The school cafeteria serves pizza every 6 school days, chicken tenders every 8 school days, and salad every 10 school days.
a.
If pizza and salad are served together today, in how many more school days will they be served together again? Show how you calculated your answer.
Pizza and salad will be served together again 30 school days from today. I determined this by calculating the least common multiple of 6 and 10, which is 30. b.
How could you mark your calendar to know when pizza and salad would be served together on the same school day?
I would mark the calendar for every 30 school days, because 30 is the LCM and
I know that means I will get the next common multiple every 30 days. I need to make sure to mark every 30 school days because if I mark every 30 days, some days will be on the weekend or holidays when there is no school.
c.
Kevin likes chicken tenders with salad. If chicken tenders and salad are served together today, when will they be served together on the same school day again?
Explain your reasoning.
Kevin will be able to eat chicken tenders and salad in 40 school days because the least common multiple of 8 and 10 is 40. d.
If pizza, chicken tenders, and salad are all served today, when will they all be served together on the same school day again? Explain your reasoning.
The LCM of 6, 8, and 10 is 120, so the three will be served together again every
120 school days.
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t How are least common multiples helpful in determining a school’s class schedules?
t What is another situation when determining the least common multiple would be helpful.
12. The students at Independence Middle School have different schedules for sixth, seventh, and eighth grades. The sixth grade students have gym every 3 school days.
The seventh grade students have art every 4 school days, and the eighth grade students have music every 6 school days. The middle school teachers are planning a special workshop, and they have to determine when students have these subjects on the same day.
a.
If today the sixth grade has gym, the seventh grade has art, and the eighth grade has music, in how many school days will all three classes occur on the same day again?
All three classes will occur on the same day in 12 school days because 12 is the
LCM of 3, 4, and 6.
b.
If today the sixth grade has gym, the seventh grade has art, and the eighth grade has music, will these classes occur together 30 school days from today? Why or why not?
No. The three classes will not occur together 30 school days from today because 30 is not a common multiple of 3, 4, and 6.
c.
On the first day of school, the sixth grade has gym, the seventh grade has art, and the eighth grade has music. If the school year is 180 school days, how many times during the school year will the three classes be held on the same day?
Since all three classes will occur on the same day every 12 school days, the occurrence of all three classes taking place on the same day throughout the school year should happen 15 times in the school year. To calculate how many times all three classes will occur on the same day during the school year, I took the number of school days in the school year (180) and divided it by the least common multiple of all three classes
(12), which results in a quotient of 15.
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Can the least common multiple be determined for any group of numbers?
13. Determine the least common multiple for each pair of numbers using prime factorization. Create factor trees if you need help determining the prime factorization of each number.
a.
8 and 12
8 5 2 3
12 5 2 2 ?
3
LCM 5 2 3 ?
3 5 24 b.
126 and 45
126 5 2 ?
3 ?
7
45 5 3 2 ?
5
LCM 5 2 ?
3 2 ?
5 ?
7 5 630 c.
165 and 22
165 5 3 ?
5 ?
11
22 5 2 ?
11
LCM 5 2 ?
3 ?
5 ?
11 5 330 d.
6, 7, and 45
6 5 2 3 3
7 5 1 3 7
45 5 3 2 3 5
LCM 5 2 3 3 2 3 5 3 7 5 630
Students explain how to determine the least common multiple for any two numbers in their own words.
Have students work in pairs to complete Questions 1 and 2.
Then share the responses as a class.
1. Why do you think the least common multiple is not always the product of the two numbers?
If the two numbers share a common factor, then their least common multiple will not be their product.
2. Explain how you can determine the least common multiple of any two numbers.
I can list the multiples of each number until I determine the smallest multiple they both have in common. I can also calculate the least common multiple of any two numbers by calculating the prime factorization of each number. The least common multiple will be the product of the prime factors the greatest number of times they appear in either prime factorization.
Be prepared to share your solutions and methods.
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