Precalculus Examples

8 x y

$8 x y$ and

12 x

$12 x$

Step 1

Since

8 x y, 12 x

$8 x y, 12 x$ contains both numbers and variables, there are two steps to find the GCF (HCF). Find GCF for the numeric part, then find GCF for the variable part.

Steps to find the GCF for

8 x y, 12 x

$8 x y, 12 x$ :

1. Find the GCF for the numerical part

8, 12

$8, 12$

2. Find the GCF for the variable part

x^{1}, y^{1}, x^{1}

$x^{1}, y^{1}, x^{1}$

3. Multiply the values together

Step 2

Find the common factors for the numerical part:

8, 12

$8, 12$

Step 3

The factors for

8

$8$ are

1, 2, 4, 8

$1, 2, 4, 8$ .

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Step 3.1

The factors for

8

$8$ are all numbers between

1

$1$ and

8

$8$ , which divide

8

$8$ evenly.

Check numbers between

1

$1$ and

8

$8$

Step 3.2

Find the factor pairs of

8

$8$ where

x \cdot y = 8

$x \cdot y = 8$ .

\begin{matrix} x & y 1 & 8 2 & 4 \end{matrix}

$\begin{array}{c} x & y \\ 1 & 8 \\ 2 & 4 \end{array}$

Step 3.3

List the factors for

8

$8$ .

1, 2, 4, 8

$1, 2, 4, 8$

1, 2, 4, 8

$1, 2, 4, 8$

Step 4

The factors for

12

$12$ are

1, 2, 3, 4, 6, 12

$1, 2, 3, 4, 6, 12$ .

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Step 4.1

The factors for

12

$12$ are all numbers between

1

$1$ and

12

$12$ , which divide

12

$12$ evenly.

Check numbers between

1

$1$ and

12

$12$

Step 4.2

Find the factor pairs of

12

$12$ where

x \cdot y = 12

$x \cdot y = 12$ .

\begin{matrix} x & y 1 & 12 2 & 6 3 & 4 \end{matrix}

$\begin{array}{c} x & y \\ 1 & 12 \\ 2 & 6 \\ 3 & 4 \end{array}$

Step 4.3

List the factors for

12

$12$ .

1, 2, 3, 4, 6, 12

$1, 2, 3, 4, 6, 12$

1, 2, 3, 4, 6, 12

$1, 2, 3, 4, 6, 12$

Step 5

List all the factors for

8, 12

$8, 12$ to find the common factors.

8

$8$ :

1, 2, 4, 8

$1, 2, 4, 8$

12

$12$ :

1, 2, 3, 4, 6, 12

$1, 2, 3, 4, 6, 12$

Step 6

The common factors for

8, 12

$8, 12$ are

1, 2, 4

$1, 2, 4$ .

1, 2, 4

$1, 2, 4$

Step 7

The GCF for the numerical part is

4

$4$ .

{GCF}_{Numerical} = 4

${GCF}_{Numerical} = 4$

Step 8

Next, find the common factors for the variable part:

x,y,x

Step 9

The factor for

x^{1}

$x^{1}$ is

x

$x$ itself.

x

$x$

Step 10

The factor for

y^{1}

$y^{1}$ is

y

$y$ itself.

y

$y$

Step 11

The factor for

x^{1}

$x^{1}$ is

x

$x$ itself.

x

$x$

Step 12

List all the factors for

x^{1}, y^{1}, x^{1}

$x^{1}, y^{1}, x^{1}$ to find the common factors.

x^{1} = x

$x^{1} = x$

y^{1} = y

$y^{1} = y$

x^{1} = x

$x^{1} = x$

Step 13

The common factor for the variables

x^{1}, y^{1}, x^{1}

$x^{1}, y^{1}, x^{1}$ is

x

$x$ .

x

$x$

Step 14

The GCF for the variable part is

x

$x$ .

{GCF}_{Variable} = x

${GCF}_{Variable} = x$

Step 15

Multiply the GCF of the numerical part

4

$4$ and the GCF of the variable part

$x$ .

$4 x$