Greatest Common Factor

What Is the GCF?

The greatest common factor (GCF) is the largest number that divides evenly into two or more numbers. Think of it as the biggest number that all your numbers share as a factor.

For example, the GCF of 12 and 18 is 6, because 6 is the largest number that goes into both 12 and 18 evenly. Sure, 1, 2, and 3 also divide into both numbers, but 6 is the greatest.

Finding the GCF is essential for simplifying fractions, factoring polynomials, and solving many algebra problems. It's one of those foundational skills that comes up over and over.

Listing Method

The most straightforward way to find the GCF is to list all the factors of each number and find the largest one they share.

Example 1: Find the GCF of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18

Common factors: 1, 2, 3, 6

The greatest common factor is 6.

Example 2: Find the GCF of 24 and 36.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Common factors: 1, 2, 3, 4, 6, 12

The greatest common factor is 12.

This method works great for small numbers, but gets tedious when the numbers are large. That's where prime factorization comes in.

Prime Factorization Method

This method is more efficient for larger numbers. The idea is to break each number down into its prime factors, then take all the common prime factors using the smallest exponent.

Example 3: Find the GCF of 48 and 72.

First, find the prime factorization of each:

$48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3$
$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$

Now identify the common prime factors. Both have 2's and 3's. Take the smallest power of each:

Both have 2 as a factor. The smallest power is $2^3$.
Both have 3 as a factor. The smallest power is $3^1$.

Multiply these together: $$\text{GCF} = 2^3 \times 3 = 8 \times 3 = 24$$

Example 4: Find the GCF of 180 and 234.

Prime factorization:

$180 = 2^2 \times 3^2 \times 5$
$234 = 2 \times 3^2 \times 13$

Common factors:

Both have $2$ (smallest power: $2^1$)
Both have $3^2$

$$\text{GCF} = 2 \times 3^2 = 2 \times 9 = 18$$

Example 5: Find the GCF of 50 and 35.

Prime factorization:

$50 = 2 \times 5^2$
$35 = 5 \times 7$

Common factor: Both have one 5.

$$\text{GCF} = 5$$

GCF with Variables

When you're working with algebraic expressions, you need to find the GCF of both the numbers and the variables.

For variables, take the one with the smallest exponent that appears in all terms.

Example 6: Find the GCF of $12x^3$ and $18x^2$.

First, find the GCF of the coefficients (12 and 18): $$\text{GCF of 12 and 18} = 6$$

Now look at the variables. Both have $x$, but the smallest power is $x^2$.

$$\text{GCF} = 6x^2$$

Example 7: Find the GCF of $24a^3b^2$ and $36a^2b^4$.

GCF of coefficients (24 and 36):

$24 = 2^3 \times 3$
$36 = 2^2 \times 3^2$

$$\text{GCF} = 2^2 \times 3 = 12$$

For variables:

Both have $a$: smallest power is $a^2$
Both have $b$: smallest power is $b^2$

$$\text{GCF} = 12a^2b^2$$

Example 8: Find the GCF of $15x^2y$, $25xy^3$, and $20x^3y^2$.

GCF of coefficients (15, 25, 20):

$15 = 3 \times 5$
$25 = 5^2$
$20 = 2^2 \times 5$

Common factor: 5

For variables:

All have $x$: smallest power is $x^1$ (or just $x$)
All have $y$: smallest power is $y^1$ (or just $y$)

$$\text{GCF} = 5xy$$

Why It Matters

The GCF shows up constantly in algebra, and here's why it's so important:

Simplifying fractions: To reduce $\frac{48}{72}$ to lowest terms, divide both numerator and denominator by their GCF (24): $\frac{48 \div 24}{72 \div 24} = \frac{2}{3}$.

Factoring polynomials: When you factor $6x^2 + 9x$, you first pull out the GCF ($3x$): $3x(2x + 3)$. This is the first step in many factoring problems.

Solving equations: Many equations are easier to solve after factoring out the GCF. The equation $6x^2 + 9x = 0$ becomes $3x(2x + 3) = 0$, which is much easier to solve.

Real-world problems: If you're arranging items into equal groups, the GCF tells you the largest group size possible. For example, if you have 24 apples and 36 oranges and want to make identical gift baskets, the GCF (12) tells you that you can make 12 baskets, each with 2 apples and 3 oranges.

Practice Problems

Try these on your own, then check your answers.

Find the GCF of 18 and 27
Find the GCF of 42 and 63
Find the GCF of 100 and 150
Find the GCF of $16x^4$ and $24x^2$
Find the GCF of $30a^2b^3$ and $45a^3b$
Find the GCF of 28, 42, and 56
Simplify $\frac{48}{60}$ by dividing by the GCF
Factor out the GCF: $12x^3 + 18x^2$

Check Your Work

9
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 27: 1, 3, 9, 27
21
$42 = 2 \times 3 \times 7$, $63 = 3^2 \times 7$
GCF = $3 \times 7 = 21$
50
$100 = 2^2 \times 5^2$, $150 = 2 \times 3 \times 5^2$
GCF = $2 \times 5^2 = 50$
$8x^2$
GCF of 16 and 24 is 8
Smallest power of $x$ is $x^2$
$15a^2b$
GCF of 30 and 45 is 15
Smallest powers: $a^2$ and $b^1$
14
$28 = 2^2 \times 7$, $42 = 2 \times 3 \times 7$, $56 = 2^3 \times 7$
GCF = $2 \times 7 = 14$
$\frac{4}{5}$
GCF of 48 and 60 is 12
$\frac{48 \div 12}{60 \div 12} = \frac{4}{5}$
$6x^2(2x + 3)$
GCF is $6x^2$
$12x^3 + 18x^2 = 6x^2(2x + 3)$

Quick Tips

When finding the GCF of two numbers, if one divides evenly into the other, the smaller number is the GCF. For example, the GCF of 15 and 45 is 15.

If two numbers share no common factors except 1, their GCF is 1. We call these numbers "relatively prime" or "coprime."

The GCF is never larger than the smallest number in your set. So if you're finding the GCF of 12 and 30, you know it can't be bigger than 12.

For variables, always take the smallest exponent. If you have $x^5$ and $x^2$, the GCF includes $x^2$, not $x^5$.

The GCF divides evenly into every number in your set. That's a quick way to check your answer: does your GCF divide into all the original numbers without a remainder?