Find the greatest common factor
of these monomials. And when they say monomials,
that’s just a fancy word for saying a one term expression. Each of these only, obviously,
have one term in them. Now to find the greatest common
factor of these, the way I think about it is, I like
to break up each of these terms into their constituent
parts. Make them a product of the
simplest things possible. For regular numbers like 10, to
me that means break them up into their prime factors, and
for these variable expressions like cd squared, break it up
into the product of the most simple variable, for example,
c times d times d. So let’s do that for each of
them and see what the greatest common factor is. Where do these overlap in
terms of their factor? And we care about the
greatest overlap. So let’s do this first one. 10 cd squared, what
is that equal to? Well 10 is equal to 2 times 5. You could do a factoring tree
here, but these are pretty straightforward numbers
to factor into. They’re prime factors. So 10 is 2 times 5, c, all you
can do is break that, you could just write that as
a c, you can’t really simplify that anymore. And d squared can be written
as d times d. So I have essentially broken
10cd squared into this, into the product of kind of the
smallest constituents that I could think of. The prime factors of 10,
and then c, and then d. Now let’s do 5cd. Well 5cd, 5 is prime, so its
prime factorization is literally just 5. c you can’t break that down
anymore, that’s just a c, and then times a d. So we really didn’t do anything
to this expression right there. And then finally you have 25c
to the third d squared. Well 25 is 5 times 5, and then
we have times c times c times c, that’s what c to the third
is, and then we have times d squared, times d times d. Now, what is the greatest common
factor, or what is the greatest common overlap between
these three things? Well they all have a 5. Let me circle them. You have a 5 there, you have a
5 there, you have a 5 there. They all have at least one c. You have one c there,
one c there, and then another c there. And they all have
at least one d. You have a d there, you have
a d there, and then you have a d there. Now they don’t all have a second
d, only the first one and the third one
have a second d. And they all don’t have a second
or third c, only this last one has a second
or third c. So we’re essentially done. The greatest common
factor is 5cd. In fact you can’t have a greater
number than 5cd be a common factor, because the
largest factor of 5cd is 5cd. So the greatest common factor
of these three monomials, or these three expressions,
is 5cd. The largest number of factors
that overlaps with all three of these expressions is
a 5, one c, and one d.