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To determine which of the given options are common factors of \( 6930 \) and \( 9009 \), we need to first find the prime factorization of both numbers.

### Step 1: Prime Factorization
1. **Prime factorization of \( 9009 \):**
   \[
   9009 = 3^2 \times 7 \times 11 \times 13
   \]

2. **Prime factorization of \( 6930 \):**
   Let's factor \( 6930 \) by breaking it down:
   \[
   6930 \div 2 = 3465 \quad (\text{so } 6930 = 2 \times 3465)
   \]
   Now factor \( 3465 \):
   \[
   3465 \div 3 = 1155 \quad (\text{so } 3465 = 3 \times 1155)
   \]
   Next, factor \( 1155 \):
   \[
   1155 \div 3 = 385 \quad (\text{so } 1155 = 3 \times 385)
   \]
   Now factor \( 385 \):
   \[
   385 \div 5 = 77 \quad (\text{so } 385 = 5 \times 77)
   \]
   Finally, factor \( 77 \):
   \[
   77 = 7 \times 11
   \]
   Putting this all together, the prime factorization of \( 6930 \) is:
   \[
   6930 = 2 \times 3^2 \times 5 \times 7 \times 11
   \]

### Step 2: Identify Common Prime Factors
To find the common prime factors, look for factors that are present in both factorizations:
   - \( 3^2 \) appears in both.
   - \( 7 \) and \( 11 \) also appear in both.

The common prime factorization is thus:
\[
3^2 \times 7 \times 11
\]

### Step 3: Analyze the Options
Now, let’s look at each option and see if it matches the common prime factors found:
1. \( 3^2 \times 7 \times 11 \): **Correct**
2. \( 2 \times 3 \): Not a common factor, as it lacks \( 3^2 \), \( 7 \), and \( 11 \).
3. \( 2 \times 5 \times 13 \): Not a common factor, lacks \( 3 \), \( 7 \), and \( 11 \).
4. \( 2 \times 3 \times 5 \times 7 \times 11 \times 13 \): Incorrect because it includes \( 2 \), \( 5 \), and \( 13 \), which are not part of \( 9009 \)’s factorization.
5. \( 3^3 \): Incorrect because \( 3^3 \) is not a factor of \( 9009 \) or \( 6930 \).
6. \( 3 \times 11 \): Correct, as both \( 3 \) and \( 11 \) are part of the common factors.

### Conclusion
The correct answers are:
- \( 3^2 \times 7 \times 11 \)
- \( 3 \times 11 \)

These two choices represent common factors of both \( 6930 \) and \( 9009 \).

Would you like further details on the factorization process, or do you have any questions about this solution?

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### Additional Questions:
1. How can you determine if a number is prime?
2. What is the difference between a factor and a prime factor?
3. Can a composite number have multiple sets of prime factors?
4. How do you find the Greatest Common Divisor (GCD) using prime factors?
5. Why is prime factorization useful in mathematics?

**Tip:** Prime factorization can help simplify fractions and find least common multiples (LCM) and greatest common divisors (GCD) easily.

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

Learn how to determine the common factors of 6930 and 9009 by using their prime factor decompositions. This problem teaches key concepts in number theory, such as prime factorization and finding the greatest common divisor (GCD).

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