5.4: Least Common Multiple
Introduction
The Decoration Committee
As the sixth grade has been planning for the social, each cluster formed a decoration committee. Each decoration committee was given the opportunity every few days to meet in the art room and make decorations for the social. Some students worked on banners, some worked on posters, some worked with streamers. All of the students had a terrific time.
The big conflict is that every few days both groups seem to be in the art room at the same time and there are never enough supplies for everyone. Mr. Caron the art teacher wants to figure out why this keeps happening.
Cluster 6A gets to work in the art room every two days.
Cluster 6B gets to work in the art room every three days.
If Mr. Caron could figure out when the groups are both in the art room on the same day, then he would have more art supplies ready. Or on those days, he could plan for the students to work on a bigger project.
If 6A works in the art room every two days and 6B works in the art room every three days, when is the first day that all of the students will be working in the art room together?
This problem may seem challenging to figure out, but if you use multiples and least common multiples, you will be able to help Mr. Caron figure out the schedule.
Pay attention and at the end of the lesson you will help solve the dilemma.
What You Will Learn
In this lesson, you will learn to:
- Find common multiples of different numbers.
- Find the least common multiple of given numbers using lists.
- Find the least common multiple of given numbers using prime factorization.
- Find two numbers given the greatest common factor and the least common multiple.
Teaching Time
I. Find the Common Multiples of Different Numbers
In mathematics, you have been working with multiples for a long time. One of the first things that you probably learned was how to count by twos or threes. Counting by twos and threes is counting by multiples. When you were small, you didn’t call it “counting by multiples,” but that is exactly what you were doing.
What is a multiple?
A multiple is the product of a quantity and a whole number.
What does that mean exactly?
It means that when you take a number like 3 that becomes the quantity. Then you multiply that quantity times different whole numbers.
3 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 6, 3 \begin{align*}\times\end{align*} 3 \begin{align*}=\end{align*} 9, 3 \begin{align*}\times\end{align*} 4 \begin{align*}=\end{align*} 12, 3 \begin{align*}\times\end{align*} 5 \begin{align*}=\end{align*} 15, 3 \begin{align*}\times\end{align*} 6 \begin{align*}=\end{align*} 18
Listing out these products is the same as listing out multiples.
3, 6, 9, 12, 15, 18.....
You can see that this is also the same as counting by threes.
The dots at the end mean that these multiples can go on and on and on. Each numbers has an infinite number of multiples.
Example
List five the multiples for 4.
To do this, we can think of taking the quantity 4 and multiplying it by 2, 3, 4, 5, 6.....
4 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 8, 4 \begin{align*}\times\end{align*} 3 \begin{align*}=\end{align*} 12, 4 \begin{align*}\times\end{align*} 4 \begin{align*}=\end{align*} 16, 4 \begin{align*}\times\end{align*} 5 \begin{align*}=\end{align*} 20, 4 \begin{align*}\times\end{align*} 6 \begin{align*}=\end{align*} 24
Our answer is 8, 12, 16, 20, 24....
Notice that we could keep on listing multiples of 4 forever.
What is a common multiple?
A common multiple is a multiple that two or more numbers have in common.
Example
What are the common multiples of 3 and 4?
To start to find the common multiples, we first need to write out the multiples for 3 and 4. To find the most common multiples that we can, we can list out multiples through multiplying by 12.
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48
The common multiples of 3 and 4 are 12, 24, 36.
Now it is time for you to practice a few.
- List out five multiples of 6.
- List out five multiples of 8.
- What are the common multiples of 6 and 8?
Take a few minutes to check your work with a peer.
II. Find the Least Common Multiple of Given Numbers Using Lists
We can also find the least common multiple of a pair of numbers.
What is the least common multiple?
The least common multiple (LCM) is just what it sounds like, the smallest multiple that two numbers have in common.
Let’s look back at the common multiples for 3 and 4.
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48
Here we know that the common multiples are 12, 24 and 36.
The LCM of these two numbers is 12. It is the smallest number that they both have in common.
We used lists of multiples for 3 and 4 to find the common multiples and then the least common multiple.
Find the Least Common Multiple for each pair of numbers.
- 5 and 3
- 2 and 6
- 4 and 6
Check your answers with your neighbor. Did you find the correct LCM?
III. Find the Least Common Multiple of Given Numbers Using Prime Factorization
Remember back to factoring numbers? We worked on using factor trees to factor numbers or to break down numbers into their primes. Take a look at this one.
\begin{align*}& \ \ \ \quad 12\\ & \ \quad \ \big / \ \ \big\backslash\\ & \quad \ 4 \quad \ \ 3\\ & \quad \big / \ \big\backslash\\ & \ \ 2 \ \ \ 2\\ & 2^2 \times 3\end{align*}
We used a factor tree in this example to factor twelve down to the prime factors of 2 squared times 3.
We can also use prime factorization when looking for the least common multiple.
How can we use prime factorization to find the LCM?
If we wanted to find the LCM of two numbers without listing out all of the multiples, we could do it by using prime factorization.
Example
What is the LCM of 9 and 12?
First, we factor both numbers to their primes.
\begin{align*}& \quad \ \ 9 && \quad \ 12\\ & \quad \big / \ \ \big\backslash && \quad \big / \ \ \big\backslash\\ & \ 3 \qquad 3 && \ 3 \qquad 4\\ &&& \ \qquad \big / \ \big\backslash\\ &&& \qquad 2 \quad \ 2\end{align*}
Next, we identify any shared primes. With 9 and 12, 3 is a shared prime number.
Then, we take the shared prime and multiply it with all of the other prime factors.
3 \begin{align*}\times\end{align*} 3 \begin{align*}\times\end{align*} 2 \begin{align*}\times\end{align*} 2
The first 3 is the shared prime factor.
The other numbers are the other prime factors.
Our answer is 36. The LCM of 9 and 12 is 36.
Now it is time for you to try.
1. Find the LCM of 4 and 10 using prime factorization.
Take a minute to check your work with a peer.
Before moving on take a few notes on multiples and finding the LCM.
IV. Find Two Numbers Given the GCF and the LCM
This section is a bit more advanced than some of the work that we have been doing. We are going to be playing detective. A detective is someone who uses clues to figure something out.
The task that you will have as a detective is to figure out two missing numbers if you have only been given the greatest common factor and the least common multiple.
If you were given the least common multiple of 10, you could think of possible numbers that would multiply to equal 10.
2 would be a possibility for one of the numbers since 2 \begin{align*}\times\end{align*} 5 \begin{align*}=\end{align*} 10.
5 would be another possibility for one of the numbers since 5 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 10.
This one was easier to figure out because the numbers are small. We didn’t even need to know the greatest common factor.
What do we do when the numbers aren’t small?
When working with larger numbers, we can use a formula to figure out missing parts.
\begin{align*}GCF(LCM) = ab\end{align*}
The GCF times the LCM is equal to number \begin{align*}a\end{align*} times number \begin{align*}b\end{align*}.
Remember that \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are variables that represent unknown numbers.
Now let’s apply this formula with an example.
Example
GCF is 6. LCM = 36.
If one of the missing numbers is 12, can you find the other missing number?
First, we take our known quantities and put them into the formula.
\begin{align*}6(36) = 12b\end{align*}
Next, we multiply the left side of the equation.
\begin{align*}216 = 12b\end{align*}
To solve for \begin{align*}b\end{align*}, we can ask ourselves, “What number times 12 is 216?” Said another way, we can divide 216 by 12.
\begin{align*}& \overset{\quad \ \overset{18} {\underline{\;\;\;\;\;\;\;\;\;\;}}}{12 \big ) 216}\end{align*}
Our answer is that \begin{align*}b\end{align*} is 18.
Real Life Example Completed
The Decoration Committee
Now that you have learned all about least common multiples, it is time to help Mr. Caron with the decoration committees.
Here is the problem once again.
As the sixth grade has been planning for the social, each cluster formed a decoration committee. Each decoration committee was given the opportunity every few days to meet in the art room and make decorations for the social. Some students worked on banners, some worked on posters, some worked with streamers. All of the students had a terrific time.
The big conflict is that every few days both groups seem to be in the art room at the same time and there are never enough supplies for everyone. Mr. Caron, the art teacher, wants to figure out why this keeps happening.
Cluster 6A gets to work in the art room every two days.
Cluster 6B gets to work in the art room every three days.
If Mr. Caron could figure out when the groups are both in the art room on the same day, then he would have more art supplies ready. Or on those days, he could plan for the students to work on a bigger project.
If 6A works in the art room every two days and 6B works in the art room every three days, when is the first day that all of the students will be working in the art room together?
First, let’s underline the important question that we are trying to solve.
Next, let’s think about how to solve this dilemma. We want to know the first common day that both 6A and 6B will meet in the art room. If you think about this, it is the same as a least common multiple.
Since 6A meets every two days, two will be the first quantity.
Since 6B meets every three days, three will be the second quantity.
Now let’s list the multiples of two and three. The common multiples will show the days that the students will both meet in the art room. The least common multiple will show the first day that the students will both meet in the art room.
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
3 6 9 12 15 18 21 24 27 30
The common multiples are 6, 12, 18, 24, 30.
The least common multiple is 6. The students will both be in the art room on these days.
Expand this to think about this next question.
If the students start the decoration committee on a Monday, what is the first day of the week that the students will both be in the art room?
We can make a list of days to figure this out.
Day 1 Monday
Day 2 Tuesday
Day 3 Wednesday
Day 4 Thursday
Day 5 Friday
Day 6 Monday – this the first day that both groups will be in the art room at the same time
Sometimes when you have a scheduling conflict like the one Mr. Caron had, using least common multiples is a great way to solve it!!
Technology Integration
Khan Academy Least Common Multiple
James Sousa Least Common Multiple
James Sousa Example of Determining Least Common Multiple Using a List of Multiples
James Sousa Example of Determining Least Common Multiple Using Prime Factorization
Other Videos:
- http://www.mathplayground.com/howto_gcflcm.html – This video covers finding the greatest common factor and the least common multiple of two numbers.
- http://www.teachertube.com/members/music.php?music_id=1351&title=Mr_Duey_GCF_LCM – This is a song only, but it is a great rap about greatest common factor and least common multiple. You'll need to register at the website to access this song.
- http://www.teachertube.com/members/viewVideo.php?video_id=15601&title=LCM_and_GCF_Indian_Method – This is a different way of finding the greatest common factor and the least common multiple. You'll need to register at the website to access this video.
Time to Practice
Directions: List the first five multiples for each of the following numbers.
1. 3
2. 5
3. 6
4. 7
5. 8
Directions: Find two common multiples of each pair of numbers.
6. 3 and 5
7. 2 and 3
8. 3 and 4
9. 2 and 6
10. 3 and 9
11. 5 and 7
12. 4 and 12
13. 5 and 6
14. 10 and 12
15. 5 and 8
Directions: Go back through the common multiples for numbers 6 – 15 and select the LCM for each pair of numbers.
16. LCM =
17. LCM =
18. LCM =
19. LCM =
20. LCM =
21. LCM =
22. LCM =
23. LCM =
24. LCM =
25. LCM =