Question

finding the gcf of two monomials

which of the following terms have a gcf of \\(6p^3\\)? select two options.

• \\(12p^3r\\)
• \\(27p^4q\\)
• \\(45p^3q^6\\)
• \\(54p^3\\)
• \\(63p^3q^6\\)

Explanation:

Analyze the target GCF

Using the Greatest Common Factor of Monomials and Greatest Common Factor knowledge points

$$ \text{Target GCF} = 6p^3 $$

This means the selected monomials must both be divisible by \(6p^3\), and their greatest common factor must be exactly \(6p^3\). Thus, for each chosen monomial:

1. The coefficient must be a multiple of \(6\).
2. The power of \(p\) must be at least \(3\).
3. Any other variables (like \(q\) or \(r\)) cannot be common to both in a way that increases the GCF, and the greatest common divisor of their coefficients and variable powers must resolve to exactly \(6p^3\).

Evaluate the given options

Using the Greatest Common Factor of Monomials knowledge point

$$
LATEXBLOCK0
$$

Verify the GCF of the selected pair

Using the Greatest Common Factor of Monomials and Greatest Common Factor knowledge points

$$
LATEXBLOCK1
$$

Answer:

• (A) \(12p^3r\) (Correct answer)
• (B) \(27p^4q\)
• (C) \(45p^3q^6\)
• (D) \(54p^3\) (Correct answer)
• (E) \(63p^3q^6\)