Pre-Algebra Examples

$126 a b^{3}$ , $- 54 a$ , $198 b$ , $- 90 a b$

Step 1

Since $126 a b^{3}, - 54 a, 198 b, - 90 a b$ 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 $126 a b^{3}, - 54 a, 198 b, - 90 a b$ :

1. Find the GCF for the numerical part $126, - 54, 198, - 90$

2. Find the GCF for the variable part $a^{1}, b^{3}, a^{1}, b^{1}, a^{1}, b^{1}$

3. Multiply the values together

Step 2

Find the common factors for the numerical part:

$126, - 54, 198, - 90$

Step 3

The factors for $126$ are $1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126$ .

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Step 3.1

The factors for $126$ are all numbers between $1$ and $126$ , which divide $126$ evenly.

Check numbers between $1$ and $126$

Step 3.2

Find the factor pairs of $126$ where $x \cdot y = 126$ .

$\begin{array}{c} x & y \\ 1 & 126 \\ 2 & 63 \\ 3 & 42 \\ 6 & 21 \\ 7 & 18 \\ 9 & 14 \end{array}$

Step 3.3

List the factors for $126$ .

$1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126$

Step 4

The factors for $- 54$ are $1, 2, 3, 6, 9, 18, 27, 54$ .

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Step 4.1

The factors for $- 54$ are all numbers between $1$ and $54$ , which divide $- 54$ evenly.

Check numbers between $1$ and $54$

Step 4.2

Find the factor pairs of $- 54$ where $x \cdot y = - 54$ .

$\begin{array}{c} x & y \\ 1 & 54 \\ 2 & 27 \\ 3 & 18 \\ 6 & 9 \end{array}$

Step 4.3

List the factors for $- 54$ .

$1, 2, 3, 6, 9, 18, 27, 54$

Step 5

The factors for $198$ are $1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198$ .

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Step 5.1

The factors for $198$ are all numbers between $1$ and $198$ , which divide $198$ evenly.

Check numbers between $1$ and $198$

Step 5.2

Find the factor pairs of $198$ where $x \cdot y = 198$ .

$\begin{array}{c} x & y \\ 1 & 198 \\ 2 & 99 \\ 3 & 66 \\ 6 & 33 \\ 9 & 22 \\ 11 & 18 \end{array}$

Step 5.3

List the factors for $198$ .

$1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198$

Step 6

The factors for $- 90$ are $1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90$ .

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Step 6.1

The factors for $- 90$ are all numbers between $1$ and $90$ , which divide $- 90$ evenly.

Check numbers between $1$ and $90$

Step 6.2

Find the factor pairs of $- 90$ where $x \cdot y = - 90$ .

$\begin{array}{c} x & y \\ 1 & 90 \\ 2 & 45 \\ 3 & 30 \\ 5 & 18 \\ 6 & 15 \\ 9 & 10 \end{array}$

Step 6.3

List the factors for $- 90$ .

$1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90$

Step 7

List all the factors for $126, - 54, 198, - 90$ to find the common factors.

$126$ : $1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126$

$- 54$ : $1, 2, 3, 6, 9, 18, 27, 54$

$198$ : $1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198$

$- 90$ : $1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90$

Step 8

The common factors for $126, - 54, 198, - 90$ are $1, 2, 3, 6, 9, 18$ .

$1, 2, 3, 6, 9, 18$

Step 9

The numbers do not contain any common variable factors. The GCF (HCF) of the numerical factors $1, 2, 3, 6, 9, 18$ is $18$ .

$18$