Which subset of \( A \) has a GCF of the elements that is 3 ? A. \( \{9 y, 12 x y\} \) B. \( \{3 x, 15 x, 12 x y\} \) C. \( \{3 x, 9 y, 12 x y, 15 x\} \) D. \( \{5 x y, 15 x, 3 x\} \)

To determine which subset of \( A \) has a greatest common factor (GCF) of the elements that is 3, we will analyze each option step by step. ### Option A: \( \{9y, 12xy\} \) 1. **Find the GCF**: - The factors of \( 9y \) are \( 3^2 \cdot y \). - The factors of \( 12xy \) are \( 3 \cdot 2^2 \cdot x \cdot y \). - The GCF is determined by taking the lowest power of each common factor: - For \( 3 \): minimum is \( 3^1 \). - For \( y \): minimum is \( y^1 \). - Thus, GCF = \( 3 \). ### Option B: \( \{3x, 15x, 12xy\} \) 1. **Find the GCF**: - The factors of \( 3x \) are \( 3 \cdot x \). - The factors of \( 15x \) are \( 3 \cdot 5 \cdot x \). - The factors of \( 12xy \) are \( 3 \cdot 2^2 \cdot x \cdot y \). - The GCF is: - For \( 3 \): minimum is \( 3^1 \). - For \( x \): minimum is \( x^1 \). - Thus, GCF = \( 3x \) (not equal to 3). ### Option C: \( \{3x, 9y, 12xy, 15x\} \) 1. **Find the GCF**: - The factors of \( 3x \) are \( 3 \cdot x \). - The factors of \( 9y \) are \( 3^2 \cdot y \). - The factors of \( 12xy \) are \( 3 \cdot 2^2 \cdot x \cdot y \). - The factors of \( 15x \) are \( 3 \cdot 5 \cdot x \). - The GCF is: - For \( 3 \): minimum is \( 3^1 \). - For \( x \): minimum is \( x^0 \) (since \( 9y \) has no \( x \)). - Thus, GCF = \( 3 \) (this is equal to 3). ### Option D: \( \{5xy, 15x, 3x\} \) 1. **Find the GCF**: - The factors of \( 5xy \) are \( 5 \cdot x \cdot y \). - The factors of \( 15x \) are \( 3 \cdot 5 \cdot x \). - The factors of \( 3x \) are \( 3 \cdot x \). - The GCF is: - For \( 3 \): minimum is \( 3^1 \). - For \( 5 \): minimum is \( 5^0 \) (since \( 3x \) has no \( 5 \)). - For \( x \): minimum is \( x^1 \). - Thus, GCF = \( 3x \) (not equal to 3). ### Conclusion The only subset that has a GCF of 3 is **Option C: \( \{3x, 9y, 12xy, 15x\} \)**.