### Video Transcript

In this video, we’ll explore how to
use different strategies to find prime factorization using exponents. This is sometimes called writing a
number as a product of its prime factors and can be hugely useful in helping us to
calculate the greatest common factor or least common multiple of two or more
numbers.

Let’s begin by recalling what we
mean by a factor. A factor is a number that divides
into another number without leaving a remainder. When listing factors of a number,
we often list them in pairs for ease. For example, the product of one and
12 is 12. So a factor pair of 12 is one and
12. We also know that two times six is
12 and three times four is 12. So all the factors of 12 are one,
12, two, six, three, and four. And what about a prime number? A prime number is a number that has
exactly two factors, one and itself. The first few prime numbers that
you should be able to recognize are two, three, five, seven, 11, 13, 17, and 19. Now, a common mistake is to think
that nine is a prime number. But its factors are one, three, and
nine. It has three factors, not two.

Similarly, one is not a prime
number, since it only has one factor, itself. When we write a number as a product
of its prime factors, that’s called prime factorization, we look to find all the
prime numbers that multiply together to make the original number. There are a couple of methods. We can draw a prime factor tree or
use the division method. We’ll begin by taking a look at the
factor tree method.

Which of the following is the prime
factorization of 18? Is it (A) 18, (B) two times three
times three, (C) two, (D) two times nine, or (E) three?

When we find the prime
factorization of 18, we’re looking for all the prime numbers which multiply together
to make 18. And we know the first few prime
numbers to be two, three, five, seven, 11, and 13. It can be nice to begin by testing
whether our smallest prime number, whether two, is a factor of the original
number. Well, 18 is an even number. So we know two is definitely a
factor. In fact, 18 divided by two is
nine. And our factor tree begins. Its first two branches are two and
nine. Now, since two is a prime number,
we circle it. And this indicates we stop at this
branch here.

Nine, however, is not a prime
number. So we repeat this process for
nine. Nine is not even, so it’s not
divisible by two. And instead, we go to the next
prime on our list. Three is a factor of nine. In fact, three squared is nine. So nine divided by three is
three. This means the next two branches on
our tree are three and three. Three is prime, so we circle both
of these. And we see that we can go no
further. We aren’t done, though. We know that these three prime
numbers multiply together to make the original. So in other words, 18 is the
product of them. It’s two times three times three,
which is option (B).

Notice that we can also write this
as two times three squared. Had the question asked us to write
our original number in exponent form, then this is what we would have needed to have
done. Be careful, though, a common
mistake is just to think that we can give our answers in a list, that’s two, three,
three, or as a sum, two plus three plus three. Well, both of these are incorrect
answers. We must give the solution as a
product.

In our next example, we’ll consider
an alternative method. It’s called the division
method.

Write 60 as a product of its prime
factors, giving the answer in exponent form.

Writing a number as a product of
its prime factors is sometimes called prime factorization. Product means time. So we want all the prime numbers
which multiply together to make here the original number that’s 60. We’re going to use the division
method. Now, the first step in this method
is to find the smallest prime number that’s also a factor of the original number, so
a factor of 60. Our first few primes are two,
three, five, seven, and 11. Now, since 60 is an even number, we
know that it must be divisible by two. It has a factor of two. And so we divide 60 by this
number. 60 divided by two is 30. That’s equivalent to saying 60 is
equal to two times 30.

Our next step is really
similar. This time, though, we divide the
quotient by the smallest prime number that’s a factor of it. Once again, 30 is an even number,
so we can divide it by two. And when we do, we find that 30
divided by two is 15. This is equivalent to saying that
30 is equal to two times 15. So we replace 30 with two times
15. And we see that we’ve now written
60 as two times two times 15. We continue to repeat this process
until our quotient, the number we get after dividing, is a prime number.

15 is not divisible by two, so we
move to the next smallest prime number. That’s three. And when we divide 15 by three, we
get five. Now, that is a prime number. So we finished. We replace 15 with its prime
factors, with three and five. And we find that 60 is two times
two times three times five.

We’re not quite done, though. The question asks us to write this
in exponent form. Another way of writing two times
two is writing two squared. Remember, when we square a number,
we times it by itself. And so in exponent form, 60 is two
squared times three times five.

In our next example, we’ll look at
the prime factorization of a much larger number, this time going back to using a
factor tree.

Write the prime factorization of
392 in exponent form.

The prime factorization of 392 is
the product of all the prime numbers, which multiply together to make 392. And it’s always sensible to list
out the first few prime numbers. They are two, three, five, seven,
11, and 13. We’re going to draw a prime factor
tree to work out the prime factorization of 392. We begin by finding a factor of
392. A really easy one is to check
whether the number is even. If it’s even, then it has a factor
of two. 392 is even, which means we can
divide it by two. And when we do, we get a result of
196. This means that, on our factor
tree, the first two branches are two and 196. Remember, though, we said two is a
prime number. That means we circle the number
two. And it indicates to us that on this
branch at least, we stop here.

Next, we look for a factor of
196. Well, 196 is also even. So it must have a factor of
two. In fact, when we divide 196 by two,
we get 98. And that means the next two
branches in our tree are two and 98. We pop a circle around this two to
show that it’s the end of this branch, and we carry on with the 98. 98 can also be divided by two to
give us 49. And so our next two branches are
two and 49. Circling the two, we look for a
factor of 49. We know 49 in fact is divisible by
seven. And since 49 divided by seven is
seven, our next two branches are both seven. And since seven is a prime number,
we circle both of these. And we’re done with our factor
tree.

We now know that 392 is the product
of all these prime numbers, everything in the circle. It’s two times two times two times
seven times seven. But the question told us to give
our answer in exponent form. Two times two times two is two
cubed, and seven times seven is seven squared. So 392 can be written as two cubed
times seven squared.

Now, it’s important to realize that
had we chosen a different starting factor of 392, we would’ve still achieved the
same answer. For example, imagine we’d first
spotted that 392 is the same as 49 times eight. Neither of these are prime numbers,
so we’re going to perform the usual process on both branches. We write 49 as seven times seven,
which means that these two branches both have a seven. Seven is a prime, so we circle them
and finish here. We might then spot that eight is
the same as four times two. We circle the two, but four is not
prime, so we continue. Four can be written as two times
two. So our next two branches are both
two, and we circle them because they’re prime and finish here. Once again, we have two times two
times two times seven times seven.

Let’s now have a look at a prime
factorization of a larger number using the division method.

Find the prime factorization of
792.

We’re going to use the division
method to find the prime factorization of this number. When we perform this process, we
want to find all the prime numbers that multiply together to make 792. And so we list the first few prime
numbers. And then the first step is to find
the smallest prime number that’s also a factor of the original number. 792 is even, so we know it has a
factor of two. Let’s divide then 792 by two. 792 divided by two is 396. Now, we can equivalently say that
792 is equal to two times 396.

Step two is to repeat this process
with the quotient, the number we got when we performed the division. So we need to find the smallest
prime factor of 396. Well, once again, it’s even, so
that’s two. We’re, therefore, going to
calculate 396 divided by two. That’s 198. This means 396 is equal to two
times 198. And it means, in turn, we can
rewrite 792 as two times two times 198.

We continue to divide our quotients
by their smallest prime factor. And we only stop when our quotient
is also prime. So we need to divide 198 by its
smallest prime factor, which is once again two. 198 divided by two is 99. So equivalently, we can say that
198 must be equal to two times 99. We now need to divide 99 by its
smallest prime factor. But 99 is not even. So it’s not divisible by two. We do, however, know that it’s
divisible by three. And of course, we might recall that
we can check for divisibility by three by adding the digits of the number
together. If their sum is divisible by three,
then the original number is also divisible by three. In this case, nine plus nine is
18. Now, 18 is divisible by three. So 99 must be divisible by
three.

And in fact, when we divide 99 by
three, we get 33. So we rewrite 99 is three times
33. And 792 is now two times two times
two times three times 33. The smallest prime factor of 33,
the smallest factor that’s also a prime number, is three. And when we divide 33 by three, we
get 11. Now, 11 itself is also a prime
number. It’s in our list here, so we finish
dividing. We write 33 is three times 11, and
this means 792 is two times two times two times three times three times 11. We can go even further and write
this in exponent form. Two times two times two is two
cubed and three times three is three squared. We can use a dot instead of a
multiplication symbol, and we see that 792 is two cubed times three squared times
11.

In this video, we’ve learned that
writing a number as a product of its prime factors, also called prime factorization,
is finding all the prime numbers that multiply together to make the original
number. We also learned that we absolutely
must write these as a product, not a list or a sum. So 30 is two times three times
five. We also saw that we have a couple
of methods that we can use, and these are called the prime factor tree method or the
division method.