# Greatest common factor of monomials

CCSS Math: HSF.IF.C.8
Learn how to find the GCF (greatest common factor) of two monomials or more.

### What you should be familiar with before this lesson

A monomial is an expression that is the product of constants and nonnegative integer powers of $x$, like $3x^2$. A polynomial is a sum of monomials.
You can write the complete factorization of a monomial by writing the prime factorization of the coefficient and expanding the variable part. Check out our Factoring monomials article if this is new to you.

### What you will learn in this lesson

In this lesson, you will learn about the greatest common factor (GCF) and how to find this for monomials.

## Review: Greatest common factors in integers

The greatest common factor of two numbers is the greatest integer that is a factor of both numbers. For example, the GCF of $12$ and $18$ is $6$.
We can find the GCF for any two numbers by examining their prime factorizations:
• $12=\blueD{2}\cdot 2\cdot \goldD{3}$
• $18=\blueD{2}\cdot \goldD3\cdot 3$
Notice that $12$ and $18$ have a factor of $\blueD{2}$ and a factor of $\goldD{3}$ in common, and so the greatest common factor of $12$ and $18$ is $\blueD{2}\cdot \goldD{3}=6$.

## Greatest common factors in monomials

The process is similar when you are asked to find the greatest common factor of two or more monomials.
Simply write the complete factorization of each monomial and find the common factors. The product of all the common factors will be the GCF.
For example, let's find the greatest common factor of $10x^3$ and $4x$:
• $10x^3=\blueD{2}\cdot 5\cdot \goldD{x}\cdot x\cdot x$
• $4x=\blueD{2}\cdot 2\cdot \goldD{x}$
Notice that $10x^3$ and $4x$ have one factor of $\blueD{2}$ and one factor of $\goldD{x}$ in common. Therefore, their greatest common factor is $\blueD2\cdot \goldD{x}$ or $2x$.

1) What is the greatest common factor of $9x^2$ and $6x$?

Let's factor each monomial completely. Then, we can find the factors common to both monomials and multiply them to find the GCF.
• $9x^2=\blueD{3}\cdot {3}\cdot \goldD{x}\cdot x$
• $6x=2 \cdot \blueD{3}\cdot\goldD{x}$
Notice that each monomial has one factor of $\blueD3$ and one factor of $\goldD{x}$. Therefore, the greatest common factor of the monomials is $\blueD3 \cdot \goldD{x}$ or $3x$
2) What is the greatest common factor of $12x^5$ and $8x^3$?

Let's factor each monomial completely. Then, we can find the factors common to both monomials and multiply them to find the GCF.
• $12x^5=\blueD{2}\cdot \blueD{2} \cdot {3}\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x} \cdot x \cdot x$
• $8x^3=\blueD2 \cdot \blueD{2}\cdot 2 \cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}$
Notice that each monomial has two factors of $\blueD2$ and three factors of $\goldD{x}$. Therefore, the greatest common factor of the monomials is $\blueD2 \cdot \blueD2 \cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}$ or $4x^3$.
3) What is the greatest common factor of $5x^7$, $30x^4$, and $10x^3$?

Let's factor each monomial completely. Then, we can find the factors common to all three monomials and multiply them to find the GCF.
• $5x^7=\blueD5\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}\cdot x\cdot x\cdot x\cdot x$
• $30x^4=2\cdot {3}\cdot \blueD{5}\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}\cdot x$
• $10x^3= {2}\cdot \blueD5\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}$
Notice that each monomial has one factor of $\blueD{5}$ and three factors of $\goldD{x}$. Therefore, the greatest common factor of the monomials is $\blueD5\cdot \goldD{x}\cdot \goldD{x}\cdot \goldD{x}$ or $5x^3$.

## A note on the variable part of the GCF

In general, the variable part of the GCF for any two or more monomials will be equal to the variable part of the monomial with the lowest power of $x$.
For example, consider the monomials $\blueD{6}\goldD{x^5}$ and $\blueD{4}\goldD{x^2}$:
• Since the lowest power of $x$ is $\goldD{x^2}$, that will be the variable part of the GCF.
• You could then find the GCF of $\blueD6$ and $\blueD4$, which is $\blueD2$, and multiply this by $\goldD{x^2}$ to obtain $\blueD2\goldD{x^2}$, the GCF of the monomials!
Sure! The complete factorization of each monomial is given below:
• $6x^5=\blueD2\cdot 3\cdot \goldD{x}\cdot \goldD{x}\cdot x\cdot x\cdot x$
• $4x^2=\blueD2\cdot 2\cdot \goldD{x}\cdot\goldD{x}$
Notice that the GCF is indeed $2x^2$.
This is especially helpful to understand when finding the GCF of monomials with very large powers of $x$. For example, it would be very tedious to completely factor monomials like $32x^{100}$ and $16x^{88}$!

## Challenge Problems

4*)What is the greatest common factor of $20x^{76}$ and $8x^{92}$?

Notice that $20x^{76}$ is the monomial with the lowest power of $x$. So the variable part of the GCF will be $x^{76}$.
The GCF of $20$ and $8$ is $4$.
Therefore, the GCF of $20x^{76}$ and $8x^{92}$ is $4x^{76}$.
5*) What is the greatest common factor of $40x^5y^2$ and $32x^2y^3$?

To find the greatest common factor of $40x^5y^2$ and $32x^2y^3$, let's factor each monomial completely, and see what is common.
• $40x^5y^2=\blueD2\cdot \blueD2\cdot \blueD{2}\cdot 5\cdot \goldD{x}\cdot \goldD{x}\cdot {x}\cdot x\cdot x \cdot\greenD{y}\cdot \greenD{y}$
• $32x^2y^3=\blueD{2}\cdot \blueD{2}\cdot\blueD{2}\cdot{2}\cdot {2}\cdot \goldD{x}\cdot \goldD{x}\cdot\greenD{y}\cdot \greenD{y}\cdot y$
The monomials each have three factors of $\blueD{2}$, two factors of $\goldD{x}$, and two factors of $\greenD{y}$ in common. Therefore, the greatest common factor of the polynomial is $\blueD2\cdot\blueD2\cdot \blueD 2 \cdot \goldD{x}\cdot \goldD{x}\cdot \greenD{y}\cdot \greenD{y}$ or $8x^2y^2$.
Alternatively, notice that the lowest power of $x$ is $x^2$ and the lowest power of $y$ is $y^2$. So the variable part of the GCF will be $x^2y^2$. The GCF of $40$ and $32$ is $8$, and so the GCF of the monomials is $8x^2y^2$.

## What's next?

To see how we can use these skills to factor polynomials, check out our next article on factoring out the greatest common factor!