17. Find the greatest common monomial factor. \( 45 a b^{2} c^{3} \) and \( 27 a b^{3} c^{4} \) a \( 9 a^{3} b c^{2} \) b \( 3 a b^{3} c^{2} \) c \( 9 a b^{2} c^{3} \) d \( 3 a^{2} b c^{3} \)

To find the greatest common monomial factor (GCMF) of the two expressions \( 45 a b^{2} c^{3} \) and \( 27 a b^{3} c^{4} \), we will break down each term into its prime factors and then identify the common factors. 1. **Factor the coefficients:** - \( 45 = 3^2 \times 5 \) - \( 27 = 3^3 \) The GCF of the coefficients \( 45 \) and \( 27 \) is \( 9 \) (since \( 3^2 \) is the highest power of \( 3 \) that divides both). 2. **Identify the variable factors:** - For \( a \): The minimum power of \( a \) in both terms is \( a^1 \). - For \( b \): The minimum power of \( b \) in both terms is \( b^2 \) (since \( b^2 \) is in the first term and \( b^3 \) is in the second term). - For \( c \): The minimum power of \( c \) in both terms is \( c^3 \) (since \( c^3 \) is in the first term and \( c^4 \) is in the second term). 3. **Combine the GCF of the coefficients and the variable factors:** - GCMF = \( 9 a^1 b^2 c^3 \) Thus, the greatest common monomial factor is: \[ 9 a b^{2} c^{3} \] Now, let's compare this with the provided options: a) \( 9 a^{3} b c^{2} \) b) \( 3 a b^{3} c^{2} \) c) \( 9 a b^{2} c^{3} \) d) \( 3 a^{2} b c^{3} \) The correct answer is: **c) \( 9 a b^{2} c^{3} \)**.