Creating a tree decomposition diagram is an easy way to find all the factors of a number. Once you understand how to create decomposition trees, it becomes easier to perform more complex tasks, such as finding the greatest common divisor or least common multiple.
Steps
Part 1 of 3: Creating a Factorization Tree
Step 1. Write a number at the top of the page
When you need to create a factoring tree for a certain number, you need to start by writing it at the top of the page. It will be the tip of your tree.
- Prepare the tree for its factors by drawing two oblique lines below the number, one pointing to the right, the other to the left.
- Alternatively, you can draw the number at the bottom of the page and draw the branches upwards. It is a less common method.
-
Example. Creating a Tree to Factor 315.
- …..315
- …../…\
Do a Factor Tree Step 2 Step 2. Find a couple of factors
Take any two factors of the number you are working with. To be a factor, the product of the two numbers must return the starting number.
- These factors will form the branches of the tree.
- You can choose any two factors. The end result will be the same.
- If there are no factors other than the number itself and "1", the starting number is prime and cannot be factored.
-
Example.
- …..315
- …../…\
- …5….63
Do a Factor Tree Step 3 Step 3. Break down each element into a couple of factors
Break your two factors down into other factors in turn.
- As seen above, two numbers can only be considered factors if their product results in the current value.
- Don't break down numbers that are already prime.
-
Example.
- …..315
- …../…\
- …5….63
- ………/\
- …….7…9
Do a Factor Tree Step 4 Step 4. Continue until you have nothing but prime numbers
You will have to keep breaking down the numbers you get until you have only primes. A prime number is a number that has no factors other than 1 and itself.
- Continue as long as necessary, making as many subdivisions as possible throughout the process.
- Note that there must be no "1" in your tree.
-
Example.
- …..315
- …../…\
- …5….63
- ………/\
- …….7…9
- ………../..\
- ……….3….3
Do a Factor Tree Step 5 Step 5. Identify all prime numbers
Since prime numbers can be found at different levels of the tree, you can highlight them so that you can find them more easily. Do this by highlighting them, circling them, or writing a list.
-
Example. The prime factors are: 5, 7, 3, 3
- …..315
- …../…\
- Step 5.….63
- …………/..\
-
………
Step 7.…9
- …………../..\
-
………..
Step 3
Step 3.
- An alternative way is to always take prime factors to the next level. At the end of the problem you will find them all on the last line.
-
Example.
- …..315
- …../…\
- ….5….63
- …/……/..\
- ..5….7…9
- ../…./…./..\
- 5….7…3….3
Do a Factor Tree Step 6 Step 6. Write the prime factors in the form of an equation
Typically, you will need to show your result by writing all prime factors separated by the multiplication sign.
- If the task is to find the factorization tree, this step is not necessary.
- Example. 5 * 7 * 3 * 3
Do a Factor Tree Step 7 Step 7. Check your work
Solve the new equation you just wrote. When you multiply all the primes, the product must match the starting number.
Example. 5 * 7 * 3 * 3 = 315
Part 2 of 3: Finding the Greatest Common Divider
Do a Factor Tree Step 8 Step 1. Create a factor tree for each number in the set
To find the greatest common factor (GCF) of two or more numbers, you have to start by factoring each number into prime factors. You can use the factor tree decomposition method.
- You will need to create a separate factor tree for each number.
- The process required to create a factor tree is the same as described in the section "Creating a Factor Tree"
- The GCD between different numbers is the largest common factor they possess. This number must exactly divide each number of the starting set.
-
Example. Find the MCD between 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
Do a Factor Tree Step 9 Step 2. Identify all common factors
Look at the decomposition tree. Identify the prime factors of each number, then highlight the ones that are on both lists
- If there are no common factors in the lists, the GCD corresponds to 1.
- Example. As mentioned earlier, the factors of 195 are 3, 5, and 13; the factors of 260 are 2, 2, 5, and 13. The common factors between the two numbers are 5 and 13.
Do a Factor Tree Step 10 Step 3. Multiply the common factors together
When the numbers in the starting set have more than one prime factor in common, you have to multiply these factors together to find the GCD.
- If there is only one factor in common, that already corresponds with the MCD.
-
Example. The common factors between 195 and 260 are 5 and 13. The product of 5 times 13 is 65.
5 * 13 = 65
Do a Factor Tree Step 11 Step 4. Write your answer
The problem is over and you are ready to answer.
- You can check by dividing the starting numbers by the MCD; if that doesn't divide them exactly you must have made some mistake, otherwise the result should be correct.
-
Example The MCD of 195 and 260 is 65.
- 195 / 65 = 3
- 260 / 65 = 4
Part 3 of 3: Finding the Least Common Multiple
Do a Factor Tree Step 12 Step 1. Create a factor tree for each number in the set
To find the least common multiple (MCM) of two or more numbers, you have to prime the numbers of the problem into prime factors. Do this using the decomposition tree method.
- Create a separate factor tree for each problem number using the method described in the "Creating a Factor Tree" section.
- A multiple is a number of which the starting number is a factor. The mcm is the smallest number that is a multiple of all the numbers in the set.
-
Example. Find the mcm between 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.
Do a Factor Tree Step 13 Step 2. Find the common factors
Consider the prime factors of the starting numbers and highlight those that are common.
- Note that if you are working with more than two numbers the common factors can be shared between even two of the starting numbers, they do not need to be all factors.
- Match the common factors. To start, if a number has "2" as a factor once and another number has "2" as a factor twice, you need to count one of the "2" as a pair; the remaining "2" from the second 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.
Do a Factor Tree Step 14 Step 3. Multiply the shared factors by the unshared ones
Once you have set aside the set of shared factors, multiply them by the unshared factors of all trees.
- The shared factors can be considered as one number. The factors you do not share must all be considered, even if they are repeated several times.
-
Example. The common factor is 5. The number 15 also contributes the unshared factor 3, and the number 40 also contributes the unshared factors 2, 2, and 2. So, you have to multiply:
5 * 3 * 2 * 2 * 2 = 120
Do a Factor Tree Step 15 Step 4. Write your answer
This completes the problem, so you should be able to write the final solution.
