We want to determine the factors of one hundred eighty.
The factors are the whole numbers that divide evenly into
another whole number. So we want all of the whole numbers
that divide evenly into one hundred eighty. It's also true that if
x times y equals one hundred eighty, and x and y are whole numbers,
then x and y are factors of one hundred eighty.
So instead of trying to think of all the whole numbers that divide evenly into one hundred eighty,
I think it's often easier to use this definition of a factor.
But we'll also use the prime factorization of one hundred eighty
to help us determine all of the factors of one hundred eighty.
So we'll start by making a prime factorization tree for one hundred eighty
So we can break down one hundred eighty as ten times eighteen,
and then ten breaks down into two times five,
which are both prime. We can write eighteen as three times six,
three is prime. And we can write six as two times three, which are both prime
So the prime factorization of one hundred eighty
would be two factors of two, two factors of three,
and one factor of five.
Now we'll make a multiplication table for one hundred eighty,
and in doing so, we'll determine the factors of one hundred eighty.
So the two most obvious whole numbers that have a product of one hundred eighty,
would be one times one hundred eighty,
so these are both factors of one hundred eighty.
And now looking at the prime factorization of one hundred eighty,
we could write one hundred eighty as two times
two, times three, times three, times five,
which would be two times, this would be ninety,
which would give us two more factors of one hundred eighty.
And again, looking at the prime factorization, we can see that three
would be a factor of one hundred eighty.
So we'd have three times, we'd have two times two, times three, times five,
which would be sixty.
Again, this gives us two more factors of
one hundred eighty. And again looking at the prime factorization we can see that four is
also a factor of
one hundred eighty since two times two is equal to four.
So here we'd have four times three, times three, times five,
that's going to be forty-five. So four times forty-five is equal to one hundred eighty.
Again which tells us that four and forty-five are factors of one hundred eighty.
Going back to our prime factors we can see that five
would be a factor of one hundred eighty. And one hundred eighty would be equal to five times
two times two, times three, times three, which would be thirty-six
So five times thirty-six
is equal to one hundred eighty. Notice we also have a factor of two times three, which would
be six.
And we can write one hundred eighty as six times
two, times three, times five, which would be thirty,
so we have six times thirty. And every time we do this we're finding two more
factors of one hundred eighty.
And also notice that I'm trying to do this in order from least to greatest,
to make sure I don't miss any factors. Notice that seven and eight
would not be factors of one hundred eighty, but nine would be.
Three times three would be nine. And we can write one hundred eighty as nine times
two, times two, times five, which would be twenty.
So nine times twenty equals one hundred eighty.
Notic ten is also a factor
of one hundred eighty here and here. And that's actually how we started our prime factorization tree.
So we can write one hundred eighty as ten times eighteen. Eleven is not a factor of
one hundred eighty, but twelve is. Because here we have two times two, times three,
which is equal to twelve.
And here we have three times five, which is equal to fifteen.
So we can write one hundred eighty as twelve times fifteen.
And now I think we have all of the factors, because notice after twelve, the next factor
would be fifteen,
which we already have listed here. So here we have a list of all the factors of one hundred eighty
The factors are one, two, three, four,
five, six, nine, ten, twelve,
fifteen, eighteen, twenty, thirty,
thirty-six, forty-five, sixty, ninety and one hundred eighty.
So you can see how was a little extra work to write out the prime factorization of one hundred eighty,
but without this, we could have easily missed some of these factors.
So the last step in this type of problem is to write out the factors
as a list from least to greatest.
So the factors of one hundred eighty
would be this entire list.
I hope you found this helpful.