5.2: Greatest Common Factors
Introduction
The Basketball Game
The sixth grade teachers have decided to have a big basketball tournament as part of the sixth grade social. The sixth graders in clusters 6A and 6B love basketball, and when the agenda is announced, all of the students are very excited.
The biggest question is how many teams to divide the students from each cluster into. The teachers want to have the same number of teams, otherwise it will be difficult to have even games for a tournament.
Cluster 6A has 48 students in it.
Cluster 6B has 44 students in it.
The teachers pose the dilemma to the students and Maria volunteers to figure out the teams.
She needs to figure out how many teams to divide each cluster into and how many students will then be on each team.
Maria has an idea how to do it. She knows that factors are going to be important. She just isn’t sure how to make certain that each cluster is divided into the same number of teams.
You can help Maria with this dilemma by learning about Greatest Common Factors, commonly called GCF’s.
Pay close attention! At the end of the lesson you will be able to help Maria with the teams.
What You Will Learn
In this lesson you will learn to complete:
- Find the greatest common factor of two or more numbers using lists.
- Find the greatest common factor of two or more numbers using factor trees.
- Solve real-world problems involving greatest common factors.
Teaching Time
I. Find the Greatest Common Factor of Two or More Numbers Using Lists
In this lesson, you will be learning about the greatest common factor (GCF).
What is the greatest common factor?
The greatest common factor is the greatest factor that two or more numbers have in common.
One way to find the GCF is to make lists of the factors for two numbers and then choose the greatest factor that the two factors have in common.
Example
Find the GCF for 12 and 16.
First, we list the factors of 12 and 16.
\begin{align*}&12 && 16\\ &12 \times 1 && 16 \times 1\\ &2 \times 6 && 8 \times 2\\ & \underline{4} \times 3 && \underline{4} \times 4\end{align*}
Next, we can underline the GCF, the largest number that appears in both lists.
The GCF is 4.
Now it is your turn to practice finding the GCF using a list. Make a list for each pair of numbers and then find the GCF of each pair.
- 24 and 36
- 10 and 18
- 18 and 45
Take a minute to check your lists with a neighbor. Did you select the correct GCF?
II. Find the Greatest Common Factor of Two or More Numbers Using Factor Trees
You just learned how to find the GCF by making lists. We can also find the GCF by making a factor tree. Let’s look at an example.
Example
Find the GCF of 20 and 30.
First, we make a factor tree for each number.
\begin{align*} & \ \ \ 20 && \quad \ \ 30\\ & \ \ \big / \ \big\backslash && \quad \ \big / \ \ \big\backslash\\ & \ 4 \quad 5 && \quad 5 \ \quad 6\\ & \big / \ \big\backslash && \qquad \ \big / \ \big\backslash\\ & 2 \ \ 2 && \qquad 3 \ \ \ 2\\ & 2^2 \times 5 && 5 \times 3 \times 2\end{align*}
Here is a tricky one because there is more than one common prime factor. We have both five and two as common factors.
When you have more than one common factor, we multiply the common factors to find the GCF.
2 \begin{align*}\times\end{align*} 5 \begin{align*}=\end{align*} 10
10 is the greatest common factor (GCF).
Stop and take a few notes on how to find the GCF of two numbers.
Now it is your turn. Use factor trees to find the GCF of each pair of numbers.
- 14 and 28
- 12 and 24
- 16 and 18
Take a minute to check your work.
Real Life Example Completed
The Basketball Game
Now that you know how to find the greatest common factor, dividing up the sixth grade clusters into teams should be a snap.
Here is the problem once again.
The sixth grade teachers have decided to have a big basketball tournament as part of the sixth grade social. The sixth graders in clusters 6A and 6B love basketball, and when the agenda is announced, all of the students are very excited.
The biggest question is how many teams to divide the students from each cluster into. The teachers want to have the same number of teams, otherwise it will be difficult to have even games for a tournament.
Cluster 6A has 48 students in it.
Cluster 6B has 44 students in it.
The teachers pose the dilemma to the students and Maria volunteers to figure out the teams.
She needs to figure out how many teams to divide each cluster into and how many students will then be on each team.
Maria has an idea how to do it. She knows that factors are going to be important. She just isn’t sure how to make certain that each cluster is divided into the same number of teams.
First, let’s underline all of the important information.
We can use the greatest common factor for the 6A and 6B to find the number of teams for each cluster.
\begin{align*}&6A = 48 && 6B = 44\\ &48 \times 1 && 44 \times 1\\ &24 \times 2 && 22 \times 2\\ &12 \times \underline{4} && 11 \times \underline{4}\\ &6 \times 8\end{align*}
The GCF of 48 and 44 is 4. The clusters can each be divided into 4 teams.
How many students will be on each team?
6A - 48 \begin{align*}\div\end{align*} 4 \begin{align*}=\end{align*} 12 students on each team
6B - 44 \begin{align*}\div\end{align*} 4 \begin{align*}=\end{align*} 11 students on each team
Now that we know about the teams, the students are ready to practice for the big basketball game!
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Factor
- a number multiplied by another number to get a product.
- Greatest Common Factor
- the greatest factor that two or more numbers has in common.
- Product
- the answer of a multiplication problem
Technology Integration
Khan Academy Greatest Common Divisor
James Sousa Greatest Common Factor
James Sousa Example of Determining the Greatest Common Factor
Other Videos:
- http://www.mathplayground.com/howto_WPGFC.html – This video goes through solving a word problem that requires finding the greatest common factor or GCF.
- http://www.mathplayground.com/howto_gcflcm.html – This video goes through finding the greatest common factor and the least common multiple of two numbers. This is a good preview for future work.
Time to Practice
Directions: Find the GCF for each pair of numbers.
1. 9 and 21
2. 4 and 16
3. 6 and 8
4. 12 and 22
5. 24 and 30
6. 35 and 47
7. 35 and 50
8. 44 and 121
9. 48 and 144
10. 60 and 75
11. 21 and 13
12. 14 and 35
13. 81 and 36
14. 90 and 80
15. 22 and 33
16. 11 and 13
17. 15 and 30
18. 28 and 63
19. 67 and 14
20. 18 and 36