Question

numbers and variables
find the gcf of the following terms. show your work.

1. 2x and 10
2. x² and 5x
3. 4x³ and 8x²
4. 12x² and 9x
5. 3x² and 8x⁵
6. 20 and x³

ready for more?
find the gcf of the following terms. show your work.
e1) 12x²y³, 30y⁴, 42x⁴y³
e2) 64cat, 32hat, 24rat

Explanation:

10) Step1: Factor each term

$2x = 2 \times x$, $10 = 2 \times 5$

10) Step2: Identify common factors

Common factor: $2$
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11) Step1: Factor each term

$x^2 = x \times x$, $5x = 5 \times x$

11) Step2: Identify common factors

Common factor: $x$
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12) Step1: Factor each term

$4x^3 = 2^2 \times x^3$, $8x^2 = 2^3 \times x^2$

12) Step2: Take lowest powers of factors

$\text{GCF} = 2^2 \times x^2 = 4x^2$
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13) Step1: Factor each term

$12x^2 = 2^2 \times 3 \times x^2$, $9x = 3^2 \times x$

13) Step2: Take lowest powers of factors

$\text{GCF} = 3 \times x = 3x$
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14) Step1: Factor each term

$3x^2 = 3 \times x^2$, $8x^5 = 2^3 \times x^5$

14) Step2: Take lowest powers of factors

$\text{GCF} = x^2$
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15) Step1: Factor each term

$20 = 2^2 \times 5$, $x^3 = x^3$

15) Step2: Identify common factors

No shared prime/variable factors, so $\text{GCF}=1$
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E1) Step1: Factor each term

$12x^2y^3 = 2^2 \times 3 \times x^2 \times y^3$, $30y^4 = 2 \times 3 \times 5 \times y^4$, $42x^4y^3 = 2 \times 3 \times 7 \times x^4 \times y^3$

E1) Step2: Take lowest powers of factors

$\text{GCF} = 2 \times 3 \times y^3 = 6y^3$
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E2) Step1: Factor each term

$64\text{cat} = 2^6 \times \text{cat}$, $32\text{hat} = 2^5 \times \text{hat}$, $24\text{rat} = 2^3 \times 3 \times \text{rat}$

E2) Step2: Take lowest powers of common factors

Common factor: $2^3 = 8$; no shared variable terms, so $\text{GCF}=8$

Answer:

1. $\boldsymbol{2}$
2. $\boldsymbol{x}$
3. $\boldsymbol{4x^2}$
4. $\boldsymbol{3x}$
5. $\boldsymbol{x^2}$
6. $\boldsymbol{1}$

E1) $\boldsymbol{6y^3}$
E2) $\boldsymbol{8}$