Find the GCF 51x^5y^3 , 120xy^2 , 84x^2y^2

$51 x^{5} y^{3}$ , $120 x y^{2}$ , $84 x^{2} y^{2}$

Step 1

Since $51 x^{5} y^{3}, 120 x y^{2}, 84 x^{2} y^{2}$ 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 $51 x^{5} y^{3}, 120 x y^{2}, 84 x^{2} y^{2}$ :

1. Find the GCF for the numerical part $51, 120, 84$

2. Find the GCF for the variable part $x^{5}, y^{3}, x^{1}, y^{2}, x^{2}, y^{2}$

3. Multiply the values together

Step 2

Find the common factors for the numerical part:

$51, 120, 84$

Step 3

The factors for $51$ are $1, 3, 17, 51$ .

The factors for $51$ are all numbers between $1$ and $51$ , which divide $51$ evenly.

Check numbers between $1$ and $51$

Find the factor pairs of $51$ where $x \cdot y = 51$ .

$\begin{array}{c} x & y \\ 1 & 51 \\ 3 & 17 \end{array}$

List the factors for $51$ .

$1, 3, 17, 51$

Step 4

The factors for $120$ are $1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120$ .

The factors for $120$ are all numbers between $1$ and $120$ , which divide $120$ evenly.

Check numbers between $1$ and $120$

Find the factor pairs of $120$ where $x \cdot y = 120$ .

$\begin{array}{c} x & y \\ 1 & 120 \\ 2 & 60 \\ 3 & 40 \\ 4 & 30 \\ 5 & 24 \\ 6 & 20 \\ 8 & 15 \\ 10 & 12 \end{array}$

List the factors for $120$ .

$1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120$

Step 5

The factors for $84$ are $1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$ .

The factors for $84$ are all numbers between $1$ and $84$ , which divide $84$ evenly.

Check numbers between $1$ and $84$

Find the factor pairs of $84$ where $x \cdot y = 84$ .

$\begin{array}{c} x & y \\ 1 & 84 \\ 2 & 42 \\ 3 & 28 \\ 4 & 21 \\ 6 & 14 \\ 7 & 12 \end{array}$

List the factors for $84$ .

$1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$

Step 6

List all the factors for $51, 120, 84$ to find the common factors.

$51$ : $1, 3, 17, 51$

$120$ : $1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120$

$84$ : $1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$

Step 7

The common factors for $51, 120, 84$ are $1, 3$ .

$1, 3$

Step 8

The GCF for the numerical part is $3$ .

${GCF}_{Numerical} = 3$

Step 9

Next, find the common factors for the variable part:

$x^{5}, y^{3}, x, y^{2}, x^{2}, y^{2}$

Step 10

The factors for $x^{5}$ are $x \cdot x \cdot x \cdot x \cdot x$ .

$x \cdot x \cdot x \cdot x \cdot x$

Step 11

The factors for $y^{3}$ are $y \cdot y \cdot y$ .

$y \cdot y \cdot y$

Step 12

The factor for $x^{1}$ is $x$ itself.

Step 13

The factors for $y^{2}$ are $y \cdot y$ .

$y \cdot y$

Step 14

The factors for $x^{2}$ are $x \cdot x$ .

$x \cdot x$

Step 15

The factors for $y^{2}$ are $y \cdot y$ .

$y \cdot y$

Step 16

List all the factors for $x^{5}, y^{3}, x^{1}, y^{2}, x^{2}, y^{2}$ to find the common factors.

$x^{5} = x \cdot x \cdot x \cdot x \cdot x$

$y^{3} = y \cdot y \cdot y$

$x^{1} = x$

$y^{2} = y \cdot y$

$x^{2} = x \cdot x$

$y^{2} = y \cdot y$

Step 17

The common factors for the variables $x^{5}, y^{3}, x^{1}, y^{2}, x^{2}, y^{2}$ are $x \cdot y \cdot y$ .

$x \cdot y \cdot y$

Step 18

The GCF for the variable part is $x y^{2}$ .

${GCF}_{Variable} = x y^{2}$

Step 19

Multiply the GCF of the numerical part $3$ and the GCF of the variable part $x y^{2}$ .

$3 x y^{2}$

Find the GCF 51x^5y^3 , 120xy^2 , 84x^2y^2

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