Let’s take a look at finding the
greatest common factor. But before we find the greatest
common factor, we probably should talk about what the greatest common factor is. To define greatest common factor,
let’s start here with the word “factor.” Factors are numbers that are
multiplied together to produce another number. For example, with the number 12,
two times six equals 12. Two and six are both factors of
12. Three and four are also factors of
It’s also possible that you’ve
heard of something called a factor tree. And that looks something like
this. We use a factor tree when we are
interested in finding prime factors. We’ll also use it today when we’re
looking at finding greatest common factor.
Okay, so now that we’ve talked
about factor, we can talk about greatest and common and then put them all
together. The word “greatest” here means just
what you think it means, the largest, the biggest, the highest in numerical
value. And the word “common” here means
shared by all the numbers that you are comparing. So when we find the greatest common
factor, we’ll be comparing two or more numbers. And we want something that is
shared by all of them.
When we put the definition
together, we could say the greatest common factor is the largest shared number that
can be divided into each number you’re comparing. We should also note that many times
we see the abbreviated form GCF. And that means greatest common
factor. Here’s an example.
Find the greatest common factor
of 12 and 30.
I’m gonna start by making a
factor tree of the number 12 and the number 30. 12 is divisible by two and
six. Two times three equals six. And now I have all the prime
factors of 12. Moving on to 30, three times 10
equals 30. Two times five equals 10. And those are the prime
Once we find the factors of 12
and 30, we can start to see common factors emerge. For example, three and three
are both factors of 12 and 30. The number two is a factor of
12 and a factor of 30. But for us, the key is that
we’re looking for the greatest common factor. We can find the greatest common
factor by taking all the shared factors and multiplying them together.
Since each number is divisible
by two and three, we multiply two times three together to get six. Six is then our greatest common
factor. So the greatest common factor
of 12 and 30 is six.
Our next example asks for the
greatest common factor of nine and 14. Following the same procedure,
let’s set up factor trees for nine and 14. Nine is divisible by three. Its prime factors are three and
three. We’re finished there. And for 14, we have two and
seven. Both of those are prime
numbers, so that’s all we can do for our factor tree.
Take a look at the factors of
nine and 14. So something is happening here
that is different from our first example. I know what you’re thinking, no
common factors? So how does this question work
if there are no common factors? But there actually is one
number that is still a factor for both nine and 14. You can multiply nine and 14 by
one. You can also divide nine and 14
by one. So the greatest common factor
here is the number one. If numbers don’t share any
other common factors, we know that they always share one.
To encourage public
transportation, Leo wants to gift some of his friends envelopes with bus tickets
and subway tickets in them. If he has 32 bus tickets and 80
subway tickets to split out equally among the envelopes and he wants to make
sure no tickets are left over, what is the greatest number of envelopes Leo can
For this problem, we’re gonna
need to use — you guessed it — the greatest common factor. The greatest common factor of
32 and 80 will help us determine exactly how many envelopes Leo can make,
because it will tell us how many groups we can divide each of these things into
We’ll start this process the
same way we always do, finding the factors of 32 and 80. 32 is divisible by two and
16. Two times eight equals 16. Two times four equals
eight. Two times two equals four. With 80, I’m gonna start with
eight and 10. Eight times 10 equals 80. Two times four equals
eight. Two times two equals four. Two times five equals 10.
Here are all of the
factors. And now let’s figure out which
ones are the common factors. Both 32 and 80 have four twos
as their factors. Two to the fourth power is the
greatest common factor of 32 and 80. So we need to find out what is
two to the fourth power. Two to the fourth power equals
16. This tells us that Leo can make
16 envelopes. Leo can then give out 16
envelopes. And in those 16 envelopes, each
envelope will have two bus passes and five subway tickets. They will all be equal. There will be seven things in
each envelope: two bus tickets, five subway tickets.
Let’s look at one other type of
This example has three numbers
that we are comparing and trying to find the greatest common factor of. We are gonna use the same
process. So no matter if there’s three
or four or five or six numbers, you’re trying to find the greatest common factor
of, you’re going to use the same procedure that we’ve been using. We’re gonna use factor trees to
find the greatest common factor.
And so here we go with our
factor trees. 10 times 11 is 110. Two times five equals 10. And now for 40, four times 10,
four equals two times two. 10 equals two times five. 12 times 10 equals 120. Two times six equals 12. Two times five equals 10. Finally, two times three equals
Each of these numbers have two
and five in common, which means their greatest common factor is 10. We can also see this in this
part of each factor tree. Two times five equals 10. 10 is the greatest common
factor of these three numbers. And now it’s your turn to go
out and try and find the greatest common factor of some numbers.