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

Greatest common factor (GCF)

The greatest common factor is the largest number that divides evenly into all the numbers you’re comparing. One way to find it is to list the factors of each number and then choose the largest factor they share.

List of factors

A quick way to find the factors of a number is to make a list. Start at $1$ and count up, writing down every number that divides the original number with no remainder (in other words, the quotient is a whole number).

Let’s list all the factors of $16$ 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

A more organized way to find factors is to draw a factor tree. If a factor list feels hard to keep track of, this method can help. A factor tree starts with the original number, then you “branch” into two factors. You keep factoring until every branch ends in a prime number, since prime numbers can’t be factored any further.

Here is a finished example.

In the tree for $16$, the first branch is $2$ and $8$ because $2\ast 8=16$. Then $8$ branches into $2$ and $4$ because $2\ast 4=8$, and $4$ branches into $2$ and $2$ because $2\ast 2=4$. You stop when the ends of the branches are all prime.

Once you’ve built both factor trees, you can find the greatest common factor by identifying the largest factor the two numbers share. For $16$ and $48$, the greatest common factor is $16$.

Least common multiple (LCM)

The least common multiple is the smallest number that both original numbers can “reach” by multiplying each one by whole numbers. You can find it by listing multiples of each number and then choosing the smallest multiple they have in common.

List of multiples

Instead of listing factors (numbers that divide into the original), you’ll list multiples (numbers you get by multiplying the original). Multiply each number by $1$, $2$, $3$, and so on.

Use the example of $12$ and $9$ to visualize a list of multiples, then find the least common multiple.

$12:12,24,36,48,60,72,...$

$9:9,18,27,36,45,54,63,72,...$

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

These lists come from multiplying each original number by $1$, $2$, $3$, and so on. Looking for numbers that appear in both lists, you can see $36$ and $72$ in common. The smaller of these is $36$, so $36$ is the least common multiple.