We want to determine the factors of 30.
Factors are whole numbers that divide evenly
into another number.
So, we're looking for all the whole numbers
that divide evenly into 30 to determine the factors of 30.
It's also true that if x times y is equal to 30,
and x and y are whole numbers,
then x and y would also be factors of 30.
So, when trying to determine the factors of a number
that's fairly small, like 30,
it's probably going to be easier if we use
this definition of a factor.
Meaning, if we can find the pairs of whole numbers
that have a product of 30, we can determine
all the factors of 30.
So, what we're going to do is make a multiplication table
for 30.
So, for example, one times 30 is equal to 30.
Therefore, one and 30 are factors of 30.
They both divide evenly into 30.
Two times 15 is also equal to 30.
So, these are factors of 30.
Next, we'd have three times ten, that's equal to 30,
and the last pair of factors would be five times six.
All of these whole numbers divide evenly into 30,
and, therefore, are factors of 30.
Now, we don't want to leave our answer like this,
we should make a list of these factors.
So, the factors of 30,
and we should list these from least to greatest.
So, we'll have one, two,
three, five,
six, ten,
15, and 30.
Now, as we try to determine the factors
of larger and larger numbers,
using a multiplication table, like this,
does become much more challenging.
So, there's also a method for determining the factors
using the prime factorization of the given number.
So, let's take a look at one more example
using the prime factorization method.
Let's try to determine the factors of 54.
We are going to make a multiplication table like we did before,
but we're going to use the prime factorization of 54
to help us complete the multiplication table.
So, let's start by listing the prime factorization of 54.
So, we'll make a factor-tree for 54.
Well, 54 is equal to six times nine.
Neither of these are prime.
Six is equal to two times three, which are both prime,
and nine is equal to three times three,
which are both prime.
So, for the prime factorization of 54,
we would have one factor of two,
and three factors of three.
So, we'll use this to help us make a complete
multiplication table for 54.
So, now we're going to try to determine
pairs of whole numbers that have a product of 54,
and we don't want to forget about the most obvious one
of one times 54.
Next, looking at the prime factorization,
54 is equal to two times
three times three times three, that would be 27.
So, two times 27 would be equal to 54.
We could also use one factor of three,
and then two times three times three,
that would be three times 18.
These would also be two factors of 54.
We can see that four and five are not factors of 54,
but six is.
Here we have two times three, that's six.
So, six times three times three, that's nine,
or six times nine.
Those would be factors of 54.
And then looking at the prime factorization of 54,
we're not going to be able to use any other combinations
of these prime factors to determine any additional factors
because there aren't any.
So, the factors of 54
are going to be one, two, three,
six, nine, 18, 27,
and 54.
Now, if you know your multiplication tables really well,
the prime factorization of 54 probably wasn't necessary
to determine all the factors of 54,
but it can be very helpful
in determining factors of much larger numbers
which we'll take a look at in the next example.
I hope this was helpful.