Part One: Making a Factor Tree
1
Write the number at the top of your paper. When you need to build a factor tree for a particular number, you need to start by writing that number at the top of the paper. This will be the tip of your 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.
Example: Make a factor tree for the number 315.
315
/...\
Find a pair of factors. Pick any pair of factors for the number you're working with. To qualify as a pair of factors, the product of the two numbers must equal your original number when multiplied together.[1]
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.
Example:
315
/...\
...563
Break down each set into its own factors. Break down your first two factors into their own sets of two factors apiece.
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.
Example:
315
/...\
...563
/ \
7...9
Repeat until you reach nothing but prime numbers. You will need to break down each number as far as possible until you separate it into nothing but prime numbers. A prime number is a number that has no factors other than itself and the number “1.”
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.
Example:
315
/...\
...563
/..\
7...9
/..\
33
Identify all of the prime numbers. Since the prime numbers might be scattered throughout various levels of the factor tree, you should identify each one to make them easier to spot. Do so by highlighting, circling, or writing them down in a list.
Example: The prime number factors are: 5, 7, 3, 3
315
/...\
...563
/..\
7...9
/..\
33
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.[2]
Example:
315
/...\
563
...//..\
..57...9
..///..\
57...33
Write out the prime factor in equation form. Typically, you would show the results of your work by writing out all of the prime number factors in a multiplication equation. Write out all of the numbers and separate each one with a multiplication sign.[3]
If you are instructed to leave your answer in factor tree form, however, this step is not necessary.
Example: 5 * 7 * 3 * 3
Check your work. Solve the new equation you just wrote. When you multiply all of the prime number factors together, the product you find should be the same as your original number.
Example: 5 * 7 * 3 * 3 = 315
