5.1: Prime Factorization
Introduction
The Sixth Grade Social
The sixth grade class is having a social in four weeks on a Friday night. The last time that the sixth grade had a social, it was a little unorganized and the teachers weren’t happy. This time, Allison (President of the sixth grade class) and Hector (the Vice President) have promised to organize it and have a plan for all of the students.
Allison and Hector have been working together to plan different activities. They have decided to have music in the gym, food in the cafeteria, board games in one classroom and basketball outside in the courtyard. They think that having enough options will keep things less chaotic.
Now that they have the activities planned, they have to figure out how to arrange the students in groups. Each group will have a certain period of time at each activity. The sixth grade has two clusters made up of two classes each.
Cluster 6A has 48 students in it.
Cluster 6B has 44 students in it.
Allison and Hector want to arrange the clusters into reasonably sized groups so that the students can hang out together, but so that the teachers will be happy too.
They are struggling with how best to arrange the students to visit each of the four activities. They want the groups to be a small enough size, but to be even too.
This is where you come in. Factors are the best way for Allison and Hector to solve this dilemma. They will also need to remember rules for divisibility to figure out how to divide up the students.
Pay close attention during this lesson, and you will know how to arrange each group of students for the sixth grade social.
What You Will Learn
In this lesson you will learn to:
- Find factor pairs of given numbers.
- Use divisibility rules to find factors of given numbers.
- Classify given numbers as prime or composite.
- Write the prime factorization of given numbers using a factor tree.
Teaching Time
I. Find Factor Pairs of Given Numbers
This lesson is all about factors, and that is where we are going to start. In order to complete the work in this lesson, you will first need to understand and identify a factor.
What is a factor?
When you multiply, the numbers that are being multiplied together are the factors of the product. Said another way, a factor is a number or a group of number that are multiplied together for a product. Groups of numbers including subtraction or addition operations are not single factors.
In this lesson, you will be finding factor pairs. This is when only two numbers are multiplied together for a product.
Let’s find some factors.
Example
What are two factors of twelve?
Here we want to find two factors of twelve or two numbers that multiply together to give us twelve. We could list many possible factors for twelve. Let’s choose 3 and 4.
Our answer is 3 \begin{align*}\times\end{align*} 4.
What if we wanted to list out all of the factors of twelve?
To do this systematically, we should first start with the number 1. Yes, one is a factor of twelve. In fact, one is a factor of every number because any number can be multiplied by one to get itself as a product.
1 \begin{align*}\times\end{align*} 12
After starting with 1, we can move on to 2, then 3 and so on until we have listed out all of the factors for 12.
\begin{align*}&1 \times 12\\ &2 \times 6\\ &3 \times 4\end{align*}
5, 7, 8 etc are not factors of 12 because we can’t multiply them by another number to get 12.
These are all of the factors for 12.
Take a few minutes to list out all of the factors for 36.
- 36
Now check your work with a peer. Did you get them all?
II. Use Divisibility Rules to Find Factors of Given Numbers
With the examples in the last section it wasn’t too difficult to find the factors for the number because we know our times tables. Sometimes, particularly with a larger number, it can be more challenging to identify the factors.
When we have a larger number that we are factoring, we may need to use divisibility rules to help us find the factors of that number.
What are divisibility rules?
Divisibility rules help determine if a number is divisible by let’s say 2 or 3 or 4. This can help us to identify the factors of a number.
Here is a chart that shows all of the basic divisibility rules.
Now some of these rules are going to be more useful than others, but you can use this chart to help you.
Example
What numbers is 1346 divisible by?
To solve this, we can go through each rule and see if it applies.
- The last digit is even-this number is divisible by 2.
- The sum of all the digits is 14-this number is not divisible by 3.
- The last two digits are not divisible by 4-this number is not divisible by 4.
- The last digit is not zero or five-this number is not divisible by 5.
- 1346 - 12 = 1334-this number is not divisible by 7.
- The last three numbers are not divisible by 8.
- The sum of the digits is 14-this number is not divisible by 9
- The number does not end in zero-this number is not divisible by 10
- The number is not divisible by 3 and 4
Our answer is that this number is divisible by 2.
Whew! That is a lot of work! You won’t usually have to go through each rule of divisibility, but it is important that you know and understand them just in case.
Try a few on your own, explain why or why not.
- Is 3450 divisible by 10?
- Is 1298 divisible by 3?
- Is 3678 divisible by 2?
Take a minute to check your work with a neighbor.
III. Classify Given Numbers as Prime or Composite
Now that you have learned all about identifying and finding factors, we can move on to organizing numbers. We can put numbers into two different categories. These categories are prime and composite. The number of factors that a number has determines whether the number is considered a prime number or a composite number.
What is a prime number?
Prime numbers are special numbers. As you can see in the text box, a prime number has only two factors. You can only multiply one and the number itself to get a prime number.
Example
Think about 13. Is it a prime number?
Yes. You can only get thirteen if you multiply 1 and 13. Therefore it is prime.
Here is a chart of prime numbers.
Be particularly careful when considering the number "1". One is neither prime nor composite.
What is a composite number?
A composite number is a number that has more than two factors. Most numbers are composite numbers. We can see from the chart that there are 25 prime numbers between 1 and 100. The rest are composite because they have more than two factors.
Take a few minutes to take some notes on prime and composite numbers.
III. Write the Prime Factorization of Given Numbers Using a Factor Tree
We can combine factoring and prime numbers together too. This is called prime factorization. When we factored numbers before, we broke down the numbers into two factors. These factors may have been prime numbers and they may have been composite numbers. It all depended on the number that we started with.
Example
Factor 36
36 can factor several different ways, but let’s say we factor it with 6 \begin{align*}\times\end{align*} 6.
These two factors are not prime factors. However, we can factor 6 and 6 again.
\begin{align*}6 &= 3 \times 2\\ 6 &= 3 \times 2\end{align*}
3 and 2 are both prime numbers.
When we factor a number all the way to its prime factors, it is called prime factorization.
It is a little tricky to keep track of all of those numbers, so we can use a factor tree to organize. Let’s organize the prime factorization of 36 into a factor tree.
Notice that at the bottom of the textbox, we wrote 36 as a product of its primes.
Is there any easier way to write this?
Yes, we can use exponents for repeated factors. If you don’t have any repeated factors, you just leave your answer alone.
\begin{align*}2 \times 2 &= 2^2\\ 3 \times 3 &= 3^2\end{align*}
The prime factorization of 36 is \begin{align*}2^2 \times 3^2\end{align*}.
Complete the prime factorization of the following number in a factor tree.
- 81
Take a minute to check your work with a friend. Did you use exponents for repeated factors? Did you remember the multiplication sign in your answer?
Real Life Example Completed
The Sixth Grade Social
You have learned all about factors and divisibility in this lesson. It is time to go back and help Allison and Hector with their work. Let’s take a look at the original problem once again.
The sixth grade class is having a social in four weeks on a Friday night. The last time that the sixth grade had a social, it was a little unorganized and the teachers weren’t happy. This time, Allison (President of the sixth grade class) and Hector (the Vice President) have promised to organize it and have a plan for all of the students.
Allison and Hector have been working together to plan four different activities. They have decided to have music in the gym, food in the cafeteria, board games in one classroom and basketball outside in the courtyard. They think that having enough options will keep things less chaotic.
Now that they have the activities planned, they have to figure out how to arrange the students in groups. Each group will have a certain period of time at each activity. The sixth grade has two clusters made up of two classes each.
Cluster 6A has 48 students in it.
Cluster 6B has 44 students in it.
Allison and Hector want to arrange the clusters into reasonably sized groups so that the students can hang out together, but so that the teachers will be happy too.
They are struggling with how best to arrange the students to visit each of the four activities. They want the groups to be a small enough size, but to be even too.
First, let’s underline the important information.
Next, let’s look at what we need to figure out. Hector and Allison need to organize the students into four groups to go with the four different activities.
They can start by writing out all of the factors for Cluster 6A. The factors will give them the combinations of students that can be sent in groups.
\begin{align*}& \ \ 48\\ &\ \ 1 \times 48\\ &\ \ 2 \times 24\\ &\ \ 3 \times 16\\ &\left . \begin{matrix} \ 4 \times 12 \\ 6 \times 8 \end{matrix} \right \} \quad \text{These are the two groups that make the most sense}\end{align*}
Now let’s find the factors of 44.
\begin{align*}&1 \times 44\\ &2 \times 22\\ &4 \times 11 - \ \text{This is the group that makes the most sense.}\end{align*}
If Hector and Allison arrange cluster 6A into 4 groups of 12 and cluster 6B into 4 groups of 11, then the groups will be about the same size.
There will be 23 students at each activity at one time. This definitely seems like a manageable number.
Allison and Hector draw out their plan. They are excited to show their plan for the evening to their teachers.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Factors
- numbers multiplied together to equal a product.
- Divisibility Rules
- a list of rules which help you to determine if a number is evenly divisible by another number.
- Prime
- a number that has two factors, one and itself.
- Composite
- a number that has more than two factors.
- Prime Factorization
- writing a number as a product of its primes.
- Factor Tree
- a diagram for organizing factors and prime factors.
Technology Integration
Khan Academy Prime Factorization
James Sousa Divisibility Rules
James Sousa Example of Prime Factorization
James Sousa Second Example of Prime Factorization
James Sousa Third Example of Prime Factorization
Other Videos:
- http://www.mathplayground.com/howto_primenumbers.html – This is a good basic video that reviews prime numbers.
- http://www.mathplayground.com/howto_primefactorization.html – This is a video on prime factorization with large numbers. It goes into more advanced prime factorization.
- http://www.mathplayground.com/howto_divisibility.html – This is a video that explains divisibility rules.
Time to Practice
Directions: List out factors for each of the following numbers.
1. 12
2. 10
3. 15
4. 16
5. 56
6. 18
7. 20
8. 22
9. 23
10. 25
11. 27
12. 31
13. 81
14. 48
15. 24
16. 30
Directions: Use what you have learned about prime and composite numbers to answer the following questions.
17. Are any of the numbers in problems 1 – 16 prime?
18. Name them.
19. What is a prime number?
20. What is composite number?
Directions: Answer each question using the rules of divisibility. Explain your answer.
21. Is 246 divisible by 2?
22. Is 393 divisible by 3?
23. Is 7450 divisible by 10?
Directions: Draw a factor tree and write 56 as a product of its primes.
24. 56
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