Greatest common factor (GCF) and least common multiple (LCM)

Greatest common factor (GCF)

The greatest common factor is the largest number that is divisible into all numbers being compared. In simpler terms, we find the greatest common factor by listing out all the factors of each number involved and indicating the largest one in common.

List of factors

The quick way to identify the factors of a number is by list. To do this, we count up from $1$ and list out all the numbers that are factors of our larger number. This means that when you divide the larger number by the factor, you get a whole number in return. Let’s list out all the factors for the two numbers $16$ and $48$, then find the greatest common factor.

$16:1(16/1=16),2(16/2=8),4(16/4=4),8(16/8=2),16(16/16=1)$

$48:1,2,3,4,6,12,16,24$

What is the greatest common factor of $16$ and $48$?

Factor tree

The organized way to identify the factors is by drawing a factor tree. If making a factor list is a bit hard to keep track of, this section will be a big help. A factor tree begins with the full numbers, and we draw branches to the tree for each number by which it may be divided. Here is a finished example.

As you can see, the top of the tree is the full number. Then, we “branch off” by finding the first set of factors. For instance, the first two branches of $16$ are $2$ and $8$ because $2\ast 8=16$. Then, the same step is done for $8$ ($2\ast 4=8$) and $4$ ($2\ast 2=4$). This is done until there is a prime number on the end of the branch, since a prime number cannot be factored any further.

Now, since we have formed the entirety of the two factor trees, we can solve for the greatest common factor by locating the largest number in common. The greatest common factor between the two numbers $16$ and $48$ is $16$.

Least common multiple (LCM)

The least common multiple is the smallest number that can be achieved by multiplying each number by different values to achieve the same result. This can be done by a similar method to the greatest common factor: with a list of multiples.

List of multiples

Instead of listing out the factors that go into our full numbers, we will list out the multiples of our numbers by multiplying them in order of $1$, $2$, $3$, etc. Use the following example of numbers $12$ and $9$ for a visualization of this list of multiples, then try to find the least common multiple.

$12:12,24,36,48,60,72,\dots$

$9:9,18,27,36,45,54,63,72,\dots$

What is the least common multiple of $12$ and $9$?

Again, we made these lists by taking our original number and multiplying it by $1$, $2$, $3$, etc. Looking for numbers in common between the two lists, you can find the numbers $36$ and $72$ in common. $36$ is the least of them, so it is the least common multiple.