How to Do a Factor Tree

Making a Factor Tree

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. 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 ...../...\ ...5....63

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 ...../...\ ...5....63 ........./ \ .......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 ...../...\ ...5....63 ........./..\ .......7...9 .........../..\ ..........3....3

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 ...../...\ ...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. Example: .....315 ...../...\ ....5....63 .../....../..\ ..5....7...9 ../..../..../..\ 5....7...3....3

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. 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

Identifying the Greatest Common Factor

Create a factor tree for each number in the set. To find the greatest common factor (GCF) between two or more numbers, you need to start by breaking down each number into its prime number factors. You can use the factor tree method to do this. 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. Example: Find 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

Identify all of the common factors. Look at all of the factor trees created for your original values. Identify the prime factors of each original number, then highlight or write down all of the factor numbers that both lists have in common. 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.

Multiply the common factors together. When two or more numbers have more than one common factor between them, you must find the GCF by multiplying all of the shared factors together. If there is only one shared factor between two or more numbers, however, the GCF is simply that single shared factor. Example: The common factors between 195 and 260 are 5 and 13. The product of 5 multiplied by 13 is 65. 5 * 13 = 65

Write your answer. The problem is now complete, and you should have your answer ready. 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. Example: The greatest common factor (GCF) of 195 and 260 is 65. 195 / 65 = 3 260 / 65 = 4

Identifying the Least Common Multiple

Create a factor tree for each number in the set. To find the least common multiple (LCM) between two or more numbers, you need to break down each number in the problem set into its prime factors. Do so by using the factor tree method. 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. Example: Find the least common multiple 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.

Find the common factors. Look at all of the prime number factors of each original value. Highlight, list, or otherwise identify all of the factors that are shared among each of the factor trees. 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.

Multiply the shared factors by those that are not shared. Once you have separated out each set of shared factors, multiply the shared factor by all of the unshared factors in each tree. The shared factor is treated as a single number. The unshared factors are each counted, even if there are multiple occurrences of that digit. Example: The common factor is 5. The number 15 also contributes the unshared factor of 3, and the number 40 also contributes the unshared factors of 2, 2, and 2. As such, you must multiply: 5 * 3 * 2 * 2 * 2 = 120

Write your answer. This completes the problem, so you should be able to write down your final answer. Example: The LCM of 15 and 40 is 120.