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How to Do a Factor Tree

Last Updated: July 21, 2023 Fact Checked

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, volunteer authors worked to edit and improve it over time. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 93,420 times. Learn more...

Creating a factor tree is one simple way to find all of the prime number factors of a number. Once you know how to do factor trees, it becomes easier to perform more advanced tasks, like finding the greatest common factor or the least common multiple.

Making a Factor Tree

• Prepare the tree for its factors by drawing two downward diagonal lines beneath the number. One should point left and the other should point right.
• Alternatively, you could place the number at the bottom of the tree and draw its factor branches up and above it. This method is far less common, however.

• These factors will form the first branches of your factor tree.
• You can pick any two factors. The end result will be the same no matter which ones you start with.
• Note that if there are no factors that equal the original number when multiplied together, other than that number and the number “1,” the number is considered a prime number and cannot be made into a factor tree.

• As before, two numbers can only be considered factors if they equal the current value when multiplied together.
• Do not break down prime numbers any further.
• ........./ \
• .......7...9

• Continue as often as needed, creating as many branches as necessary in the process.
• Note that there should be no “1” anywhere in your tree.
• ........./..\
• .........../..\
• ..........3....3

• ... 5 ....63
• ............/..\
• ......... 7 ...9
• ............../..\
• ........... 3 .... 3
• An alternate way of writing out the prime factors of a factor tree is to carry each prime factor down to the next level. By the end of the problem, you can spot each prime number because each one will be in the bottom row.
• ....5....63
• .../....../..\
• ..5....7...9
• ../..../..../..\
• 5....7...3....3

• If you are instructed to leave your answer in factor tree form, however, this step is not necessary.
• Example: 5 * 7 * 3 * 3

• Example: 5 * 7 * 3 * 3 = 315

Identifying the Greatest Common Factor

• You will need to create a separate factor tree for each number.
• The process required for making a factor tree is the same as described in the “Making a Factor Tree” section.
• The GCF between two or more numbers is the largest prime number factor that is shared between all of the given numbers in the problem. This number must divide evenly into all of the original numbers in the problem.
• ....../....\
• ....5....39
• ........./....\
• .......3.....13
• The prime factors of 195 are: 3, 5, 13
• ......./.....\
• ....10.....26
• .../...\ …/..\
• .2....5...2...13
• The prime factors of 260 are: 2, 2, 5, 13

• If there are no common factors between the numbers, the GCF is the number 1.
• Example: As noted previously, the factors of 195 are 3, 5, and 13; the factors of 260 are 2, 2, 5, and 13. The common factors between both numbers are 5 and 13.

• If there is only one shared factor between two or more numbers, however, the GCF is simply that single shared factor.
• 5 * 13 = 65

• You can double-check your work, if desired, by dividing each of your original numbers by the GCF you calculated. If the GCF goes into each number evenly, the solution should be accurate.
• 195 / 65 = 3
• 260 / 65 = 4

Identifying the Least Common Multiple

• Create a separate factor tree for each number in the problem set using the method described in the "Making a Factor Tree" section.
• A multiple is a value that the current number is a factor of. The LCM is the smallest value that can qualify as a shared multiple of all given numbers in the set.
• The prime factors of 15 are 3 and 5.
• ......../..\
• .......2...4
• ............/ \
• ..........2...2
• The prime factors of 40 are 5, 2, 2, and 2.

• Note that if you are working with more than two numbers, the common factors must be shared among at least two of the factor trees but do not need to appear in all of the trees.
• Pair off common factors. For instance, if one number has “2” as a factor twice and the other has “2” as a factor once, you should count the shared “2” as one pair; the remaining “2” of the first number will be counted as an unshared digit.
• Example: The factors of 15 are 3 and 5; the factors of 40 are 2, 2, 2, and 5. Among these factors, only the number 5 is shared.

• The shared factor is treated as a single number. The unshared factors are each counted, even if there are multiple occurrences of that digit.
• 5 * 3 * 2 * 2 * 2 = 120

• Example: The LCM of 15 and 40 is 120.

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• ↑ https://www.mathsisfun.com/definitions/factor-tree.html
• ↑ https://flexbooks.ck12.org/cbook/ck-12-middle-school-math-concepts-grade-6/section/5.6/primary/lesson/greatest-common-factor-using-factor-trees-msm6/
• ↑ http://www.bbc.co.uk/schools/gcsebitesize/maths/number/primefactorsrev1.shtml
• ↑ http://www.softschools.com/math/topics/gcf/
• ↑ https://virtualnerd.com/pre-algebra/factors-fractions-exponents/least-common-multiple/common-multiples/least-common-multiple-multiply-factors
• ↑ https://math.libretexts.org/Bookshelves/PreAlgebra/Book%3A_Prealgebra_(OpenStax)/02%3A_Introduction_to_the_Language_of_Algebra/2.10%3A_Prime_Factorization_and_the_Least_Common_Multiple_(Part_2)

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Factor Tree Worksheets

Welcome to our Factor Tree Worksheets page.

Here you will find our selection of worksheets involving using factor trees for factorize numbers.

We have a range of sheets at different levels of difficulty.

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On this page we have worksheets using factor trees to factorize a range of numbers.

We also have a link to our Prime Factorization Calculator which will quickly and easily show you all the prime factors of any number.

Using these factor tree worksheets will help your child to:

• use and understand factor trees;
• factorize a range of different numbers up to 100 and beyond.

These sheets are aimed at students from 6th grade and upwards.

What is a factor tree?

A factor tree is a visual way of factorizing a number and showing you all the prime factors that multiply together to make the number.

Factor trees have several rules:

• you cannot use the number 1 in a factor tree (or the tree just goes on forever)
• the two numbers underneath each rectangle must multiply together to make the number in the rectangle above.

What does a factor tree look like?

Let's have a look at a couple of examples.

Example 1) Complete the factor tree below and write down the prime factorization of 12.

• The number in the top rectangle is the number we are trying to find the factors for.
• The numbers in the circles are the prime factors that multiply together to give the number above.
• The numbers in the other rectangles are composite (non-prime) numbers that we still need to find the factors for.

We know that 12 = 2 x 6 and 12 = 3 x 4 (we cannot use 1 in a factor tree!)

We can use either of these equations - it does not matter which one.

Let's use 12 = 2 x 6. So the prime factor is 2 and the composite factor is 6.

This gives us:

Now we only have the 6 left to factorize.

We know that 6 = 1 x 6 and 6 = 2 x 3.

We cannot use 1 in a factor tree, so that leaves us with 6 = 2 x 3.

The prime factors of 12 are 2, 2, and 3.

This means that 12 = 2 x 2 x 3 (or 2 2 x 3) as a product of prime factors (prime factorization.

Example 2) Let us go back to Example 1) and factorize it in a different way.

This time we will factorize 12 as 3 x 4 instead of 2 x 6.

Now we only have the 4 left to factorize.

We know that 4 = 1 x 4 and 4 = 2 x 2.

We cannot use 1 in a factor tree, so that leaves us with 4 = 2 x 2.

The prime factors of 12 are still 2, 2, and 3.

So we still end up with 12 = 2 x 2 x 3 (or 2 2 x 3) as a product of prime factors (prime factorization.

So we still have the same answer as Example 1) but the factor trees have some different numbers in.

The Factor Tree in Example 1) has a 6, and the Factor Tree in Example 2) has a 4.

This does not matter - the important thing is that the numbers in the circles (the prime factors) are the same in both examples.

Factor Tree Walkthrough Video

This short video walkthrough shows our Factor Tree Worksheet 1 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, check out the video!

All the worksheets have answers provided. However, as in the example above, the answers are not unique - you can have several different factor trees for numbers.

The important thing is that the prime factorization of each number has to match the answer provided.

The sheets are graded with the easiest sheets first.

• Factor Tree Worksheet 1
• PDF version
• Factor Tree Worksheet 2
• Factor Tree Worksheet 3
• Factor Tree Worksheet 4
• Factor Tree Worksheet 5
• Factor Tree Worksheet 6
• Factor Tree Worksheet 7
• Factor Tree Worksheet 8

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

What is prime factorization?

To find out more about prime factorization, including how it works and to look at some worked examples, take a look at our prime factorization support page.

• What is Prime Factorization support page

Prime Factorization Worksheets

As well as factor tree worksheets, we also have a collection of prime factorization worksheets.

These worksheets are harder than the sheets on this page, and require an understanding of prime factorization.

We have prime factorization challenges and also prime factorization riddles here too.

• Prime Factorization Worksheets (harder)
• Prime Factorization Calculator

Our prime factorization calculator will help you factor any number into a product of its prime factors.

It will show the answer in exponential form as well as standard form.

You can use this calculator to check that your prime factors in your factor trees are correct.

Divisibility Rules 1-10 Chart

We have a range of charts which can help you determine whether a number between 1 and 10 is a factor of a number.

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• Greatest Common Factor Worksheets

As well as our calculator, we also have a range of greatest common factor worksheets.

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They will help students learn and practice finding the greatest common factor of numbers up to 100.

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Factor Tree Worksheets

Free PDFs give students practice in finding the prime factors

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Factors are numbers that divide evenly into another number, and a prime factor is a factor that is a prime number. A  factor tree  is a tool that breaks down any number into its prime factors. Factor trees are helpful tools for students because they provide a graphic representation of the prime factors that can divide into a given number. Factor trees are so named because once created, they look somewhat like a tree.

The worksheets below give students practice in creating factor trees. For example, the free printables list numbers such as 28, 44, 99, or 76 and ask students to create a factor tree for each. Some of the worksheets provide some of the prime factors and ask students to fill in the rest; others require students to create factor trees from scratch. In each section, the worksheet is printed first with an identical worksheet below it listing the answers to make grading easier.

Prime Factor Tree Worksheet No. 1

Find out how much students know about creating factor trees by having them complete this worksheet first. It requires students to create each factor tree from scratch.

Before having students start this worksheet, explain that when factoring the numbers, there is often more than one way to do so. It won’t matter which numbers they use because they will always end up with the same prime factors of the number. For example, the prime factors for 60 are 2, 3 and 5, as the example problem demonstrates.

Prime Factor Tree Worksheet No. 2

For this worksheet, students find the prime numbers for each number listed using a factor tree. If students are struggling, this worksheet may help them master the concept. It provides some of the factors, and students fill in the rest in provided blank spaces.

For example, in the first problem, students are asked to find the factors of the number 99. The first factor, 3, is listed for them. The students then find the other factors, such as 33 (3 x 33), which factors further into the prime numbers 3 x 3 x 11.

Prime Factor Tree Worksheet No. 3

This worksheet gives struggling students more help in mastering factor trees because some of the prime factors are provided for them. For example, the number 64 factors into 2 x 34, but students can further factor that number into prime factors of 2 x 2 x 17, because the number 34 can factor into 2 x 17.

Prime Factor Tree Worksheet No. 4

This worksheet provides some of the factors to help students create factor trees. If students are struggling, explain that the first number, 86, can only factor into 43 and 2 because both of those numerals are prime numbers. By contrast, 99 can factor into 8 x 12, which can further factor into (2 x 4) x (2 x 6), which further factors into the prime factors (2 x 2 x 2) x (2 x 3 x 2).

Prime Factor Tree Worksheet No. 5

Finish your factor tree lesson with this worksheet that also gives students some of the factors for each number. For further practice, have students complete these worksheets that let them find the prime factors of numbers without using factor trees.

• 4th Grade Math Lesson on Factor Trees
• Determining If a Number Is Prime
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Factor Tree Worksheets

A prime number is a number whose only factors are 1 and itself. Factor tree is a diagram used to find the prime factors of a natural number greater than 1. Each branch of the factor tree ends up in a prime number. Practicing factor tree worksheets will help students understand the concept of prime numbers in a pictorial way.

Benefits of Factor Tree Worksheets

Factors are parts of numbers which when multiplied together, give a larger number. Factorization is an important concept under prime numbers, out of which the easiest method to find factors is the factor tree method. Go ahead and solve the factor tree worksheets to understand the properties of prime numbers and factorization.

Download Factor Tree Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

Factor Trees Worksheets

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Help your students prepare for their Maths GCSE with this free Factor Trees worksheet of 44 questions and answers

• Section 1 of the factor trees worksheet contains 36 skills-based questions, in 3 groups to support differentiation
• Section 2 contains 4 applied factor trees questions with a mix of worded problems and deeper problem solving questions
• Section 3 contains 4 foundation and higher level GCSE exam style factor trees questions 
• Answer keys and a mark scheme for all factor trees questions are provided
• Follows variation theory with plenty of opportunities for students to work independently at their own level
• All questions created by fully qualified expert secondary maths teachers
• Suitable for GCSE maths revision for AQA, OCR and Edexcel exam boards

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Factor trees at a glance

Factor trees are used to find the prime factors of a number. Any number with two or more factors is called a composite number and can be written as a product of prime numbers. 

To find the prime factors of a number we can use a prime factor tree with the given number at the top. We find a factor pair of the number and represent these on two branches. Next we find factor pairs of each of these numbers and continue in this way until we get to prime numbers. These prime numbers are the prime factors of the original number. We can write a number as a product of its prime factors by writing these prime factors in a product calculation, using exponents where appropriate. 

Although multiple different prime factor trees could be created for a number, the final prime factorisation will always be the same. The prime factors of two or more numbers can be compared to find the lowest common multiple (LCM/least common multiple) or highest common factor (HCF/greatest common factor/GCF) of those numbers.

Looking forward, students can then progress to additional number worksheets , for example a n equivalent fractions & ordering fractions worksheet or simplifying fractions worksheet .

For more teaching and learning support on Number our GCSE maths lessons provide step by step support for all GCSE maths concepts.

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Factor Tree Method

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.

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 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 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

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.

Write 72 as a product of prime factors in index form.

We write 72 at the top of the tree, with two branches below:

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.

The prime factors of 8 are 2, 2, and 2. Make a circle around these factors.

Prime factors of 72 are 2, 2, 2, 3 and 3.

72 = 2 x 2 x 2 x 3 x 3.

In index form: 72 = 23 x 32

Given that 120 = 30 x 4, show the product of prime factors in index form.

We will write 4 and 30 at the top of the tree since these are the specified factors of 120.

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.

2, 3, and 5 are the prime factors of 30 . Draw a circle with these factors.

Prime factors of 120 are 2, 2, 2, 3 and 5.

120 = 2 x 2 x 2 x 3 x 5.

In index form:

120 = 23 x 3 x 5

In index form, write 126 as a product of prime factors.

We can make two branches and write the factors of 126. In the image below, 9 and     14 were used as factors of 126.

We continue factoring since 9 and 14 are both 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.

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

Using the factor tree method, express 32 as a power of 2. 

We will write 2 and 16 at the top of the tree.

Draw a circle around 2. 

The prime factors of 16 are 2, 2, 2, and 2. Draw a circle with these factors.

Prime factors of 16 are 2, 2, 2, 2 and 2.

32 = 2 x 2 x 2 x 2 x 2.

32 = 25 

Express 243 as a power of 3. 

At the top of the tree, we will write 3 and 81.

Draw a circle around 3. 

3, 3, 3, and 3 are the prime factors of 81. Make a circle with these factors.

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 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. 

• 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 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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Factoring with the Factor Tree

Factoring , i.e., listing all the prime factors, of an integer is a useful skill that often helps to solve math problems. For example, one of the ways to find the GCD ( Greatest Common Divisor ) or LCM ( Least Common Multiple ) of two integers is by listing all their prime factors. The GCD is then the product of all the common factors; the LCM is the product of all the remaining ones.

The applet below is a tool for finding such prime factorizations with the device known as the Factor Tree , meaning that you proceed a step at a time. At the beginning you are given a number, say N, which is to be factored and two boxes beneath it. The boxes are to be filled with two numbers, say, N 1 and N 2 - factors of N - such that N = N 1 × N 2 . (You only need to find one factor. If one is found correctly, the other is filled out automatically. (In "real life," you may use a calculator or the long division algorithm.) Note that the two numbers need not be prime. The one which is not will grow two boxes beneath it so that you'll be able to proceed with further factoring until all the prime factors have been found.

Do not be intimidated if you can't list all the factors right away. You do not have to. Just finding one factor will simplify the problem. Remember the divisibility rules . An even number is divisible by 2. A number is divisible by 3 if the sum of its digits is divisible by 3. The number that ends in 5 is divisible by 5 and the one that ends in 10 is divisible by 10.

While the prime factorization is unique (up to the order of factors), the factor tree reflects the order in which the factors were found, and is by no means unique. So you do not have to worry about finding a "correct" factor at any step. Any factor will do. Check the three examples of factoring of 108:

(There is a button at the bottom of the applet that allows you to repeat a problem and see for yourself that different factor trees still produce the same prime factorizaion for any number.)

At any time there may be several boxes to type in the factors. Choose any of them. After typing in a number Press the Enter key. How do you choose a box? You may click on it or you may press the Tab key 1 or more times to get to the box of your choice.

If you are reading this, your browser is not set to run Java applets. Try IE11 or Safari and declare the site https:///www.cut-the-knot.org as trusted in the Java setup.

( Elsewhere we use two factor trees simultaneously to find the greatest common divisor and the least common multiple of two numbers.)

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factor tree

Factor tree in math: a comprehensive guide, what is a factor tree in math.

In mathematics, a factor tree is a graphical representation used to break down a composite number into its prime factors. It provides a visual and systematic approach to finding the prime factors of a given number. By using a factor tree, we can easily identify all the prime factors and their corresponding powers.

Knowledge Points in Factor Tree and Step-by-Step Explanation

To understand factor trees, we need to be familiar with the following concepts:

• Prime Numbers: Numbers that are divisible only by 1 and themselves, such as 2, 3, 5, 7, etc.
• Composite Numbers: Numbers that have more than two factors, i.e., numbers that are not prime.
• Prime Factorization: The process of expressing a composite number as a product of its prime factors.

Let's break down the steps involved in creating a factor tree:

• Start with the given composite number at the top of the tree.
• Identify any two factors of the number and write them as branches below the number.
• Continue breaking down each factor into its prime factors until all branches end with prime numbers.
• Repeat this process for each branch until the entire tree is filled with prime numbers.

Formula or Equation for Factor Tree

There is no specific formula or equation for creating a factor tree. It is a visual representation that follows the steps mentioned above.

Applying the Factor Tree Formula or Equation

As mentioned earlier, there is no formula or equation for factor trees. Instead, we follow a systematic approach to break down composite numbers into their prime factors.

Symbol for Factor Tree

There is no specific symbol for factor trees. They are typically represented using a tree-like structure, with the composite number at the top and branches representing the factors.

Methods for Creating a Factor Tree

There are no specific methods for creating a factor tree, but here are a few tips to make the process easier:

• Start with the smallest possible factors to simplify the tree.
• Use prime numbers as factors whenever possible to ensure the tree ends with prime numbers.
• Break down each factor into its prime factors until all branches end with prime numbers.

Solved Examples on Factor Tree

Example 1: Create a factor tree for the number 36.

Example 2: Create a factor tree for the number 48.

Practice Problems on Factor Tree

• Create a factor tree for the number 72.
• Create a factor tree for the number 90.
• Create a factor tree for the number 120.

FAQ on Factor Tree

Q: What is a factor tree? A: A factor tree is a graphical representation used to break down a composite number into its prime factors.

Q: How do you create a factor tree? A: To create a factor tree, start with the given composite number, identify its factors, and continue breaking down each factor into its prime factors until all branches end with prime numbers.

Q: Why is prime factorization important? A: Prime factorization helps us understand the fundamental building blocks of a number and is crucial in various mathematical concepts, such as simplifying fractions, finding the greatest common divisor, and solving equations.

Q: Can factor trees be used for prime numbers? A: No, factor trees are used to break down composite numbers into their prime factors. Prime numbers have only two factors, 1 and the number itself, so they cannot be further broken down.

Q: Are factor trees unique for each number? A: No, factor trees are not unique. Different factor trees can be created for the same number, but they will all have the same prime factors.

Factor trees provide a systematic and visual approach to prime factorization, making it easier to understand the composition of composite numbers. By practicing with various examples and problems, you can enhance your skills in using factor trees effectively.

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Factor Tree Method – Definition, Facts, Examples | How to do Factor Tree Method?

The Factor Tree method is a simple way to break down the given numbers into prime factors. It is written in such a way that you get the exact divisibility by using the prime factors of the original number. Students who are unable to understand the concept of the Factor Tree Method can make use of this article and practice the problems. You can see questions with clear-cut solutions in the below section. Know how to solve the multiples using the Factor Tree Method from this page.

• Methods of Prime Factorization
• Worksheet on Methods of Prime Factorization
• Properties of Multiples

Factor Tree – Definition

How to do Prime Factorization using the Factor Tree Method?

The factor Tree Method is one of the simple ways to find the prime factors of the original number. There are some to step to solve the problems using a factor tree method. 1. The first step in making a factor tree is to find a pair of factors whose product is the number that we are factoring. 2. Check whether 2 is a factor of the given number. 3. Break down each set into factors. 4. Check for other prime numbers like 3, 5 factors of the original number. 5. Then split the factors until you get all the prime factors. 6. At last write the prime factors of the original number.

Factor Trees Examples

FAQs on Factor Tree Method

1. What is a Factor Tree?

One method for producing the prime factorization of a natural number is called a factor tree.

2. How to make a factor tree?

1. First write the given number 2. Find a pair of factors 3. Break down each set into its own factors. 4. Repeat until you reach nothing but prime numbers.

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Factor Tree Puzzles Inspired by Dr. Harold Reiter

By: Author Sarah Carter

Posted on Published: July 4, 2018  - Last updated: December 12, 2022

Categories Number Puzzles , Pencil & Paper Puzzles , Prime Factorization , Puzzles Related to Math Content

Yesterday, I shared about all the amazing things I learned/experienced as part of the Tulsa Math Teachers’ Circle Summer Immersion Workshop. As part of that post, I mentioned that I had created some factor tree puzzles after being inspired by the puzzles shared by Dr. Harold Reiter and the puzzle created by one of the other participants.

These puzzles involve a factor tree in which each terminating leg of the tree represents a prime number. Each variable represents a unique digit. So, if a is equal to five, then b cannot be equal to five.

I really enjoyed solving the puzzle created by another teacher in our group, so I decided I should try my hand at creating my own puzzles. 

Here are the four puzzles I ended up creating myself. In all honesty, I created way more than four puzzles. But, many of the puzzles I created turned out to have multiple solutions or mistakes. These four puzzles have been verified by both my husband and myself to have only one solution.

I decided to place circles around the prime numbers to further clarify the puzzles. 

The difficulty level definitely increases as the puzzle number increases. Puzzle 1 is the easiest. Puzzles 2 and 3 are trickier. And, Puzzle 4 is a beast. It took two people with math degrees a very long time to verify that there was one solution. So, I would NOT recommend giving puzzle 4 to students. Unless of course, there’s some logical step that makes solving it much easier than the route my husband and I ended up taking!

Free Download of Factor Tree Puzzles

Factor Tree Puzzles (PPT) (1564 downloads )

Factor Tree Puzzles (PDF) (1599 downloads )

I hope you enjoy trying your hand at this new-to-me type of puzzle!

P.S. I’m excited to say that my first factor tree puzzle was featured in Jo Morgan ‘s 91st Maths Gems post. If you’re not following her Resourceaholic blog , you are missing out!

Puzzle Solutions

I intentionally do not make answers to the printable math puzzles I share on my blog available online because I strive to provide learning experiences for my students that are non-google-able. I would like other teachers to be able to use these puzzles in their classrooms as well without the solutions being easily found on the Internet. However, I do recognize that us teachers are busy people and sometimes need to quickly reference an answer key to see if a student has solved a puzzle correctly or to see if they have interpreted the instructions properly. If you are a teacher who is using these puzzles in your classroom, please send me an email at [email protected] with information about what you teach and where you teach. I will be happy to forward an answer key to you.

More Printable Paper and Pencil Logic Puzzles

Tuesday 14th of January 2020

Wednesday 23rd of October 2019

Do you have the answer sheet posted anywhere?

Sunday 15th of September 2019

Are the answers posted anywhere?

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GCSE Maths (9-1) Prime Factors, HCF, LCM

Subject: Mathematics

Age range: 14-16

Resource type: Lesson (complete)

Last updated

8 September 2019

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Full lesson powerpoint and pdf which includes:

• Number Facts (Factors, Multiples, Prime Numbers)
• Writing a number in prime factor form
• Finding the HCF of two numbers
• Finding the HCF of three numbers
• Finding the LCM of two numbers
• Finding the LCM of three numbers
• Questions with full solutions and answers for the above
• Worded Problem-Solving HCF/LCM Walkthrough & Questions
• Using your casio calculator to produce prime factorisation of a number

Method used: Listing, circling common factors, venn diagrams

I have not included printable worksheets as all questions can be produced in exercise books, and some prime factor trees can be quite long! If you would prefer to have worksheets anyway, please ask and I will be happy to post those also.

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Type whole numbers greater than one into the circles to complete the factor trees. Remember that each branch of the tree will eventually end in a prime number. You can read instructions for more information about completing factor trees.

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If you are not familiar with factor trees I suggest you do the 'exercise' before attempting these puzzles. The following instructions explain how to do the exercise.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.

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Factoring Fundamentals Tutorials & Examples

Boost your math confidence—Master factoring techniques—Solve complex problems with ease

Integer Factorization

1 hr 5 min 21 Examples

• Introduction to Video: Prime Factorization
• What is Prime Factorization and GCF for Integers?
• Examples #1-6: Give the prime Factorization of each number
• Examples #7-12: List all the pairs of integral factors
• Examples #13-21: Find the GCF for two numbers

Greatest Common Factor

1 hr 25 min 26 Examples

• Introduction to Video: Greatest Common Factor
• Overview of Factoring Out the GCF and Reducing Fractions with Examples #1-2
• Examples #3-8: Divide by the common factor
• Examples #9-14: Find the GCF of each monomial
• Examples #15-20: Factor the polynomial by pulling out the GCF
• Examples #21-26: Factor by pulling out the GCF

Difference of Squares

1 hr 28 min 31 Examples

• Introduction to Video: Difference of Two Squares
• Steps for Factoring the Difference of Squares
• Examples #1-6: Square each monomial
• Examples #7-12: Express as a product of squares
• Examples #13-31: Factor the binomial, if possible

Trinomial Coefficient One

1 hr 15 min 17 Examples

• Introduction to Video: Factoring Trinomials
• Steps for Factoring Trinomials with Leading Coefficient of 1
• Examples #1-8: Factor the trinomial
• Examples #9-17: Factor the trinomial completely

Trinomial Coefficient Not One

1 hr 33 min 12 Examples

• Introduction to Video: Factoring Trinomials with leading coefficient greater than 1
• Steps for Factoring Trinomials with leading coefficient greater than 1
• Examples #1-4: Factor the trinomial
• Examples #5-8: Factor the trinomial completely, if possible
• Examples #9-13: Factor the trinomial completely

Factor by Grouping

1 hr 30 min 16 Examples

• Introduction to Video: Factoring by Grouping
• How to factor a trinomial using grouping method? Examples #1-2
• Factoring by Grouping Steps with Example #3
• Examples #4-7: Factor each polynomial by grouping
• Example #8: Factor the polynomial by grouping
• Examples #9-12: Factor by Grouping and Difference of Squares
• Examples #13-16: Factor completely, using more than one factoring method

Factoring Cubes

1 hr 2 min 11 Examples

• Introduction to Video: Factoring Cubes
• Factoring Sum/Difference of Cubes Formulas with Example #1
• Examples #2-7: Factor using the sum or difference of cubes
• Examples #8-11: Factor completely

Solve By Factoring

1 hr 57 min 18 Examples

• Introduction to Video: Solve by Factoring
• Steps for Solving a Polynomial Equation by Factoring
• Examples #1-17: Solve by Factoring
• Example #18: Solve by Factoring using two different factoring techniques

Factoring Word Problems

1 hr 22 min 9 Examples

• Introduction to Video: Factoring Word Problems
• Steps for Solving Factoring Word Problems with Example #1
• Examples #2-3: Finding two numbers that satisfy the given conditions
• Examples #4-5: Find the area
• Example #6: Find the dimension of a picture given area of border
• Example #7: Find the width of the picture frame
• Example #8: Find the side of a box given the surface area
• Example #9: Find time when the ball will reach a specific height

Chapter Test

1 hr 30 min 31 Examples

• Examples #1-3: Give the prime factorization
• Examples #4-5: List the pairs of integral factors
• Examples #6-21: Factor completely, if possible
• Examples #22-24: Factor completely
• Examples #25-28: Solve the polynomial equation by factoring
• Example #29: Solve the polynomial equation
• Examples #30-31: Solve the word problem by factoring