Creating a factor tree is an easy way to find all the prime numbers of a number. Once you know how to create a factor tree, you'll be able to perform complex calculations more easily, such as finding the greatest common factor (GCF) or least common multiple (LCM).
Step
Method 1 of 3: Creating a Factor Tree
Step 1. Write a number on the top of your paper
If you want to construct a factor tree for a number, start by writing the specific number on the top of the paper as the starting number. This number will be the top of the tree that you will create.
- Prepare a place to write the factor by drawing two diagonal lines downwards just below the number. One line sloping to the lower left, and the other sloping to the lower right.
- Alternatively, you can write the numbers at the bottom of the paper and then draw lines up as branches for the factors. However, this method is not commonly used.
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Example: Create a factor tree for the number 315.
- …..315
- …../…
Step 2. Find a pair of factors
Choose the factor pair for the starting number you are working with. To qualify as a factor pair, these factor numbers must equal the original number when they are multiplied.
- These two factors will form the first branch of your factor tree.
- You can choose any two numbers as factors because the end result will be the same no matter where you start.
- Keep in mind that no factor is ever the same as the original number when it's been multiplied, other than if this factor and your starting number are “1” and this number is a prime number that a factor tree can never build.
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Example:
- …..315
- …../…
- …5….63
Step 3. Break down each pair of factors again to get their respective factors
Describe the first two factors that you got earlier so that each has two factors.
- As explained earlier, two numbers can be considered a factor only if their product is equal to the number they divide.
- Prime numbers do not need to be subdivided.
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Example:
- …..315
- …../…
- …5….63
- ………/
- …….7…9
Step 4. Repeat the above steps until you get prime numbers
You must continue to divide until the result is only prime numbers i.e. numbers whose factors are only this number and "1."
- Continue as long as the result can still be divided by making the next branches.
- Keep in mind that there can't be a "1" in your factor tree.
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Example:
- …..315
- …../…
- …5….63
- ………/..
- …….7…9
- ………../..
- ……….3….3
Step 5. Identify all prime numbers
Because these primes occur at different levels in the factor tree, you should be able to identify each prime number to make it easier to find. You can color, circle, or write prime numbers that are already there.
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Example: The prime numbers that are factors of 315 are: 5, 7, 3, 3
- …..315
- …../…
- Step 5.….63
- …………/..
-
………
Step 7.…9
- …………../..
-
………..
Step 3
Step 3.
- Another way to write the prime factors of a factor tree is to write this number in the next level below it. At the end of solving the problem, you can see each of these prime factors because they will all be on the bottom row.
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Example:
- …..315
- …../…
- ….5….63
- …/……/..
- ..5….7…9
- ../…./…./..
- 5….7…3….3
Step 6. Write the prime factors in equation form
Write down all the prime factors you get -- as a result of the problems you've solved -- in multiplication form. Write down each factor by putting a timestamp between the two numbers.
- If you are asked to provide an answer in the form of a factor tree, you do not need to do the following steps.
- Example: 5 x 7 x 3 x 3
Step 7. Check your multiplication results
Solve the equation you just wrote. After you have multiplied all the prime factors, the result should be the same as the initial number.
Example: 5 x 7 x 3 x 3 = 315
Method 2 of 3: Determining the Greatest Common Factor (GCF)
Step 1. Create a factor tree for each initial number specified in the problem
To calculate the greatest common factor (GCF) of two or more numbers, start by breaking down each initial number into prime factors. You can use a factor tree for this calculation.
- Make a factor tree for each starting number.
- The steps required to create a factor tree here are the same as those described in the section “Creating a Factor Tree.”
- The GCF of two or more numbers is the largest factor obtained from the results of dividing the initial numbers that have been determined in the problem. The FPB must completely divide all the initial numbers in the problem.
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Example: Calculate the GCF of 195 and 260.
- ……195
- ……/….
- ….5….39
- ………/….
- …….3…..13
- The prime factors of 195 are: 3, 5, 13
- …….260
- ……./…..
- ….10…..26
- …/…\ …/..
- .2….5…2…13
- The prime factors of 260 are: 2, 2, 5, 13
Step 2. Find the common factors of these two numbers
Take a look at each factor tree you have created for each initial number. Determine the prime factors for each initial number, then color or write all the factors the same.
- If none of the factors are the same from the two initial numbers, it means that the GCF of these two numbers is 1.
- Example: As explained earlier, the factors of 195 are 3, 5, and 13; and the factors of 260 are 2, 2, 5, and 13. The common factors of these two numbers are 5 and 13.
Step 3. Multiply the factors by the same
If there are two or more numbers that are the same factor of these two numbers, you must multiply all the factors by the same to get the GCF.
- If there is only one common factor of two or earlier numbers, the GCF of these initial numbers is this factor.
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Example: The common factors of the numbers 195 and 260 are 5 and 13. The product of 5 times 13 is 65.
5 x 13 = 65
Step 4. Write down your answers
This question has now been answered, and you can write the final result.
- You can double-check your work, if necessary, by dividing each initial number by the GCF you have obtained. Your calculation result is correct if each initial number is divisible by GCF.
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Example: The GCF of 195 and 260 is 65.
- 195 / 65 = 3
- 260 / 65 = 4
Method 3 of 3: Determining Least Common Multiple (LCM)
Step 1. Make a factor tree of each initial number given in the problem
To find the least common multiple (LCM) of two or more numbers, you must break down each initial number in the problem into prime factors. Perform these calculations using a factor tree.
- Create a factor tree for each initial number in the problem according to the steps described in the section "Creating a Factor Tree."
- A multiple means a number that is a factor of a given initial number. The LCM is the smallest number that is the same multiple of all the initial numbers in the problem.
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Example: Find the LCM of 15 and 40.
- ….15
- …./..
- …3…5
- The prime factors of 15 are 3 and 5.
- …..40
- …./…
- …5….8
- ……../..
- …….2…4
- …………/
- ……….2…2
- The prime factors of 40 are 5, 2, 2, and 2.
Step 2. Determine the common factors
Note all the prime factors of each starting number. Color it, record it, or otherwise find all the factors that are common in each factor tree.
- Keep in mind that if you are working on a problem with more than two starting points, the same factor must be present in at least two of the factor trees, but not necessarily in all of the factor trees.
- Match the factors together. For example, if one starting number has two factors of “2” and another starting number has one factor of “2,” you would have to account for the factor “2” as a pair; and another “2” factor as an unpaired number.
- Example: The factors of 15 are 3 and 5; the factors of 40 are 2, 2, 2, and 5. Of these, only 5 appears as a common factor of these two initial numbers.
Step 3. Multiply the paired factor by the unpaired factor
After you separate the paired factors, multiply this factor by all the unpaired factors in each factor tree.
- Paired factors are considered as one factor, while unpaired factors must be taken into account all, even if this factor occurs several times in the factor tree of an initial number.
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Example: The paired factor is 5. The starting number 15 also has an unpaired factor of 3, and the starting number 40 also has an unpaired factor of 2, 2, and 2. So you have to multiply:
5 x 3 x 2 x 2 x 2 = 120
Step 4. Write down your answers
The problem has been answered, and now you can write the final result.
