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The question is asking for the prime factor decompositions that are **common factors** of the numbers 6930 and 9009. The prime factorizations of the two numbers are given as:

\[
6930 = 2 \times 3^2 \times 5 \times 7 \times 11
\]
\[
9009 = 3^2 \times 7 \times 11 \times 13
\]

### Step-by-step analysis:

- **Common prime factors** between 6930 and 9009:
  - \( 3^2 \) (both have this factor).
  - \( 7 \) (both have this factor).
  - \( 11 \) (both have this factor).

### Decomposition of common factors:
The common prime factors are \( 3^2 \times 7 \times 11 \). Hence, we need to select prime factor decompositions that contain only these common factors.

### Now checking the options:
1. \( 3^2 \times 7 \times 11 \) - Correct, this contains only the common factors.
2. \( 2 \times 3 \) - Incorrect, 2 is not a common factor.
3. \( 2 \times 5 \times 13 \) - Incorrect, 2, 5, and 13 are not common factors.
4. \( 3 \times 11 \) - Partially correct, but missing \( 3^2 \) and \( 7 \).
5. \( 2 \times 3^2 \times 5 \times 7 \times 11 \times 13 \) - Incorrect, contains extra factors (2, 5, 13).
6. \( 3^2 \) - Partially correct, but missing \( 7 \) and \( 11 \).

### Correct Answer:
- \( 3^2 \times 7 \times 11 \)

Do you need more details or have any questions?

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Here are 5 relative questions to expand on this topic:

1. How can you find the greatest common divisor (GCD) using prime factorization?
2. Why is \( 2 \) not a common factor of 6930 and 9009?
3. How does prime factorization help in finding least common multiples (LCMs)?
4. What are the differences between common factors and common multiples?
5. How do we handle numbers with repeated prime factors when finding common factors?

**Tip**: Prime factorization is the key to solving many problems involving greatest common divisors and least common multiples.

Which of the prime factor decompositions below are common factors of 6930 and 9009? Select all the correct answers.

Explore how to find the common factors between the prime factorizations of 6930 and 9009. This problem involves identifying the greatest common divisor using prime factors, perfect for students in Grades 8-10.

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