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 . . . . . /. . . \
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
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
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
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
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
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
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. Example: The common factors between 195 and 260 are 5 and 13. The product of 5 multiplied by 13 is 65. 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. Example: The greatest common factor (GCF) of 195 and 260 is 65. 195 / 65 = 3 260 / 65 = 4
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.
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. 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
Example: The LCM of 15 and 40 is 120.
