What is the greatest common factor of the following sets of monomials?
$30m^3{n}^2,45m^2{n}^4,75m^4{n}^3$

• A. $5m^2{n}^2$
• B. $15m^2{n}^2$
• C. $450m^4{n}^3$
• D. $15$

Answer: (B)

Verified step by step guidance

1. Identify the terms given in the problem. Since the problem is about finding the Greatest Common Factor (GCF) of terms, list each term clearly.
2. Factor each term into its prime factors, including both numerical coefficients and variables with their exponents. For example, express each term as a product of prime numbers and variables raised to powers.
3. Compare the factorizations of all terms to find the common factors. The GCF will include all prime factors and variables that appear in every term, with the smallest exponent found among the terms.
4. Write the GCF by multiplying the common prime factors and variables together, using the smallest exponents identified in the previous step.
5. Verify your GCF by dividing each original term by the GCF to ensure that the result is an integer or a simplified term without fractions.