We build a chart of the people who came before us when we make a family tree. They are called family trees because when each ancestor gives way to more ancestors, they tend to branch out.
A factor tree, on the other hand, shows the numbers that come together to create a larger number.
What is Factor Tree?
A factor tree is used to find prime factors of a number. It is made in the shape of a tree, with the provided number split into branches that show all the number’s factors. Specifically, the prime factorization of a number.
Before we get into factor tree method, there are key ideas to remember:
Factors are the numbers you multiply together to get another number as the product. Let us use the number 8 as an example. When we consider the numbers that we can multiply to get 8, we can choose 2 and 4. Also, 1 and 8 are factors of 8.
A composite number is a positive integer produced by multiplying two smaller positive integers. For instance, 4 is an even composite number. The factors of 4 are 1, 2, and 4.
The number 15 is an odd composite number with factors 1, 3, 5, and 15.
A prime number only has two factors: one and itself. All other prime numbers are odd numbers, only the number 2 is an even prime. The table shows the prime numbers between 1 and 100.
| 2 | 3 | 5 | 7 | 11 |
| 13 | 17 | 19 | 23 | 29 |
| 31 | 37 | 41 | 43 | 47 |
| 53 | 59 | 61 | 67 | 71 |
| 73 | 79 | 83 | 89 | 97 |
Because 1 isn’t a prime number, it does not show up in the factor tree.
How to build a factor tree?
A factor tree is a method of factorizing a number that works in the same manner that tree branches do. Every branch of the factor tree is divided into factors, and when the factors can no longer be factorized, the branches come to a stop, with the final factors circled and considered the prime factors of the given number.
Let us look at these steps in finding out the factors of a number using the factor tree method:
A factor tree can be created by factorizing a number until the prime factors are found. These factors are separated and written in the form of tree branches. Let us look at how to draw the factor tree of the numbers 18, 48, 54, 60 and 180.
Factor Tree of 18
The following steps can be used to compute the factors of 18.
Step 1: Write two factors of the given number.
The factors of 18 are 2 and 9.
Step 2: Make a circle around any prime factors.
Because 2 is a prime number, draw a circle around it.
Step 3: Repeat steps 1 and 2 for any remaining composite numbers.
We now have 3 and 3 as the prime factors of 9. Draw a circle around these factors.
Step 4: The prime factors of 18 are 2, 3, and 3.
Step 5: Thus, 18 = 2 x 3 x 3.
Alternative Solution (Factor Tree of 18)
The number 18 has other factor pair. Another option is to multiply 3 by 6.
Step 1: The factors of 18 are 3 and 6.
Step 2: Draw a circle around 3 since it is a prime number.
Step 3: The prime factors of 6 are 2 and 3.
Step 4: The prime factors of 18 are 2, 3, and 3.
Step 5: As a result, 18 = 2 x 3 x 3.
This means that a number can have many factor trees, all of which produce the same prime factor product.
Factor Tree of 48
The image below shows the factor tree of 48.
Step 1: The factors of 48 are 24 and 2.
Step 2: Because 2 is a prime number, draw a circle around it.
Step 3: We keep looking for factors for 24 because it is a composite number, and we come up with 4 and 6. 2 and 2 are the prime factors of 4, while 2 and 3 are the prime factors of 6. Make a circle around these factors.
Step 4: The prime factors of 48 are 2, 2, 2, 2 and 3.
Step 5: Hence, 48 = 2 x 2 x 2 x 2 x 3.
Alternative Solution (Factor Tree of 48)
Step 1: The factors of 48 are 6 and 8.
Step 2: We continue factoring since 6 and 8 are both composite numbers.
Step 3: The prime factors of 6 are 2 and 3. We keep obtaining the prime factors of 8, which gives us 2, 2 and 2.
Step 4: The prime factors of 48 are 2, 2, 2, 2 and 3.
Step 5: Thus, 48 = 2 x 2 x 2 x 2 x 3.
Factor Tree of 54
The image shows the factor tree of 54.
Step 1: The factors of 54 are 6 and 9.
Step 2: Because 6 and 9 are composite numbers, we continue to factor.
Step 3: The prime factors of 6 are 2 and 3, while the prime factors of 9 are 3 and 3. Draw a circle around these factors.
Step 4: The prime factors of 54 are 2, 3, 3, and 3.
Step 5: As a result, 54 = 2 x 3 x 3 x 3.
Factor Tree of 60
There are several factor pairs in the number 60. Because the number ends in a 0, selecting 6 x 10 is simple:
Step 1: The factors of 60 are 6 and 10.
Step 2: We continue to factor since both 6 and 10 are composite numbers.
Step 3: 2 and 3 are the prime factors of 6 while 5 and 2 are the prime factors of 10. Draw a circle around these factors.
Step 4: The prime factors of 60 are 2, 2, 3, and 5.
Step 5: As a result, 60 = 2 x 2 x 3 x 5.
Factor Tree of 120
The number 120 has several factor pairings. Choosing 12 x 10 is easy since the number ends in a 0:
Step 1: The factors of 120 are 10 and 12.
Step 2: We keep factoring since both 10 and 12 are composite numbers.
Step 3: The prime factors of 10 are 2 and 5, while the prime factors of 12 are 2, 2 and 3. Make a circle around these factors.
Step 4: The prime factors of 120 are 2, 2, 2, 3, and 5.
Step 5: As a result, 120 = 2 x 2 x 2 x 3 x 5.
Writing an answer in index notation
Index notation is a way of representing numbers that have been multiplied several times by themselves.
For example, in index notation, 2 x 2 x 2 can be written as 23 and read as “2 cubed” or “2 to the power of 3.” We are multiplying 2 by itself three times.
Similarly, 3 x 3 can also be written as 32, which is read as “3 squared” or “3 to the power of 2.” We are multiplying 3 by itself twice.
Let us use the earlier factor tree examples. Express the factors of 48 and 54 as product of prime factors in index notation.
As a result, 48 = 2 x 2 x 2 x 2 x 3. In index notation, this is expressed as 48 = 24 x 3. Which can be read as “the product of 3 and 2 to the power of 4”.
Hence, the number 54 can be written as 54 = 2 x 33 in index form. So, we have the “product of 2 and 3 to the power of 3”.
More Examples
Example 1
Use factor tree method to show 100 as a product of prime factors.
Step 1: Write two factors of the given number.
At the top of the tree, we write 100, with two branches below:
Step 2: Make a circle around any prime factors.
We keep factoring since both 4 and 25 are composite numbers. The prime factors of 4 are 2 and 2. Make a circle around these factors.
Step 3: Repeat steps 1 and 2 for any remaining composite numbers.
The prime factors of 25 are 5 and 5. Make a circle around these factors.
Step 4: List the prime factors from smallest to largest in increasing order.
Prime factors of 100 are 2, 2, 5, and 5.
Step 5: Write the factors of the numbers using product of primes.
100 = 2 x 2 x 5 x 5.
Example 2
Write 72 as a product of prime factors in index form.
Step 1: Write two factors of the given number.
We write 72 at the top of the tree, with two branches below:
Step 2: Make a circle around any prime factors.
We keep factoring since both 8 and 9 are composite numbers. The prime factors of 9 are 3 and 3. Draw a circle around these factors.
Step 3: Repeat steps 1 and 2 for any remaining composite numbers.
The prime factors of 8 are 2, 2, and 2. Make a circle around these factors.
Step 4: List the prime factors from smallest to largest in increasing order.
Prime factors of 72 are 2, 2, 2, 3 and 3.
Step 5: Write the factors of the numbers using product of primes.
72 = 2 x 2 x 2 x 3 x 3.
In index form: 72 = 23 x 32
Example 3
Given that 120 = 30 x 4, show the product of prime factors in index form.
Step 1: Write two factors of the given number.
We will write 4 and 30 at the top of the tree since these are the specified factors of 120.
Step 2: Make a circle around any prime factors.
We continue to factor because 4 and 30 are both composite numbers. 2 and 2 are the prime factors of 4. Make a circle around these factors.
Step 3: Repeat steps 1 and 2 for any remaining composite numbers.
2, 3, and 5 are the prime factors of 30. Draw a circle with these factors.
Step 4: List the prime factors from smallest to largest in increasing order.
Prime factors of 120 are 2, 2, 2, 3 and 5.
Step 5: Write the factors of the numbers using product of primes.
120 = 2 x 2 x 2 x 3 x 5.
In index form:
120 = 23 x 3 x 5
Example 4
In index form, write 126 as a product of prime factors.
Step 1: Write two factors of the given number.
We can make two branches and write the factors of 126. In the image below, 9 and 14 were used as factors of 126.
Step 2: Make a circle around any prime factors.
We continue factoring since 9 and 14 are both composite numbers.
Step 3: Repeat steps 1 and 2 for any remaining composite numbers.
The prime factors of 9 are 3 and 3 whereas the prime factors of 14 are 2 and 7. Draw a circle with these factors.
Step 4: List the prime factors from smallest to largest in increasing order.
Prime factors of 126 are 2, 3, 3, and 7.
Step 5: Write the factors of the number using product of primes.
We need to express the solution in index notation, so we write:
Product primes: 126 = 2 x 3 x 3 x 7
In index form: 126 = 2 x 32 x 7
Example 5
Using the factor tree method, express 32 as a power of 2.
Step 1: Write two factors of the given number.
We will write 2 and 16 at the top of the tree.
Step 2: Make a circle around any prime factors.
Draw a circle around 2.
Step 3: Repeat steps 1 and 2 for any remaining composite numbers.
The prime factors of 16 are 2, 2, 2, and 2. Draw a circle with these factors.
Step 4: List the prime factors from smallest to largest in increasing order.
Prime factors of 16 are 2, 2, 2, 2 and 2.
Step 5: Write the factors of the numbers using product of primes.
32 = 2 x 2 x 2 x 2 x 2.
In index form:
32 = 25
Example 6
Express 243 as a power of 3.
Step 1: Write two factors of the given number.
At the top of the tree, we will write 3 and 81.
Step 2: Make a circle around any prime factors.
Draw a circle around 3.
Step 3: Repeat steps 1 and 2 for any remaining composite numbers.
3, 3, 3, and 3 are the prime factors of 81. Make a circle with these factors.
Step 4: List the prime factors from smallest to largest in increasing order.
Prime factors of 243 are 3, 3, 3, 3 and 3.
Step 5: Write the factors of the number using product of primes.
243 = 3 x 3 x 3 x 3 x 3.
In index form:
243 = 35
Example 7
In the factor tree of 140, fill in the missing numbers. Find the factors of the number using product of primes and index notation.
Solution:
140 can be broken down into factors,10 and 14. The prime factors of 10 are 2 and 5, while the prime factors of 14 are 2 and 7. Hence,
- The number 14 is missing from the yellow box.
- The first red circle’s missing number is 5.
- In the second red circle, the missing number is 7.
- As the product of primes, 140 = 2 x 2 x 5 x 7.
- In index form: 140 = 22 x 5 x 7.
Here is a completed factor tree of 140.
Common Misconceptions
- Adding instead of multiplying
A common mistake when creating a factor tree for a number like 30 is to write the factors as 15 and 15. This is incorrect since 15 x 15 = 225.
There are several factor pairs for the number 30. The following pairs can be used: 2 and 15, 5 and 6, 3 and 10.
- Considering a number to be prime
Here are a few examples of numbers that are frequently misidentified as prime numbers: 1, 9, 15, 21, 27, 57, 91.
The factor tree does not include 1 because it is not a prime number.
The numbers 9, 15, 27, and 57 are not prime since these numbers are multiples of 3. That is, 9 = 3 x 3 , 15 = 5 x 3, and 57 = 19 x 3.
91, on the other hand is a multiple of 7, with 97 = 13 x 7.
To avoid this misconception, remember that:
A number is prime if it has only two factors: 1 and itself.
The number is composite if it has more than two factors.
2 is the only even prime number. 2 is one of the factors in all other even numbers.
- The final solution is not written
Do not simply finish your solution by just completing or demonstrating the factor tree method. You must write the number as a product of its factors once you finish the factor tree. If needed, use index notation to express the solution.
For example,
- The solution was incorrectly simplified
Mark a prime number once you have reached it in the factor tree. The diagram should be spaced out such that all the components can be seen clearly, and the prime factors for your answer should be circled.
Summary
- A factor tree is a specific diagram in which we identify the factors of a number, then the factors of those numbers, and so on, until we cannot factor them anymore. All prime factors of the original number are obtained in the end.
- To use a factor tree, follow these steps:
Step 1: Write two factors of the given number.
Step 2: Make a circle around any prime factors.
Step 3: Repeat steps 1 and 2 for any remaining composite numbers.
Step 4: You have completed factoring when you are left with just prime numbers. Use the circled prime numbers to write the prime factors of the given number. Always list the prime factors from smallest to largest in increasing order.
Step 5: Write the factors of the numbers using product of primes. (if necessary, in index form / notation).
As an example, here is a factor tree of 72:
- Index notation is a way to express numbers that have been multiplied by themselves several times.
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