what is the greatest common factor of 10xy^3 and 20xy

To find the greatest common factor (GCF) of \(10xy^3\) and \(20xy\), let's break down each expression into its prime factors:

### Step-by-Step Solution:
1. **Factorize Each Term:**
   - \( 10xy^3 = 2 \times 5 \times x \times y \times y \times y \).
   - \( 20xy = 2^2 \times 5 \times x \times y \).

2. **Identify Common Factors:**
   - The common prime factors between \( 10xy^3 \) and \( 20xy \) are: 
     - **Numerical Part:** \( 2 \times 5 = 10 \).
     - **Variable Part:** \( x \) and \( y \).

3. **Greatest Common Factor:**
   - Combine the common numerical and variable factors:  
     \[
     \text{GCF} = 10xy.
     \]

### Final Answer:
The greatest common factor is **\(10xy\)**.

Let me know if you need a detailed breakdown for each step or have any further questions. Here are 5 related questions to expand on this topic:

1. What is the GCF of \(15x^2y^3\) and \(25xy^2\)?
2. How do you find the GCF of polynomials?
3. What is the least common multiple (LCM) of \(10xy^3\) and \(20xy\)?
4. Can you have a GCF with negative coefficients? If so, how?
5. What is the GCF of \(30x^2y\) and \(45xy^2\)?

**Tip:** Always factorize completely when finding the GCF to ensure no common factors are overlooked.

Learn how to find the greatest common factor (GCF) of the algebraic expressions 10xy^3 and 20xy through prime factorization. This step-by-step solution explains how to break down each term and identify common factors. Suitable for Grades 7-9.

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