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To complete this table, let's break down each part systematically:

### Given Information:
The number \( N \) is given as:
\[
N = 3 \cdot 11 \cdot 17 \cdot 23
\]
So, the prime factors of \( N \) are \( 3 \), \( 11 \), \( 17 \), and \( 23 \).

### Table Completion Steps:

1. **Prime Factors of \( N \)**:
   The prime factors of \( N \) are \( 3 \), \( 11 \), \( 17 \), and \( 23 \). These fill the first row.

2. **Factors of \( N \) with Exactly Two Prime Factors**:
   These factors are products formed by choosing exactly two of the four prime factors. The possible combinations are:
   - \( 3 \cdot 11 \)
   - \( 3 \cdot 17 \)
   - \( 3 \cdot 23 \)
   - \( 11 \cdot 17 \)
   - \( 11 \cdot 23 \)
   - \( 17 \cdot 23 \)

3. **Factors of \( N \) with Exactly Three Prime Factors**:
   These factors are products formed by choosing exactly three of the four prime factors. The possible combinations are:
   - \( 3 \cdot 11 \cdot 17 \)
   - \( 3 \cdot 11 \cdot 23 \)
   - \( 3 \cdot 17 \cdot 23 \)
   - \( 11 \cdot 17 \cdot 23 \)

### Completed Table:

| Description                                  | Factor Combinations       |
|----------------------------------------------|----------------------------|
| **The prime factors of \( N \)**             | \( 3 \), \( 11 \), \( 17 \), \( 23 \) |
| **Factors with exactly two prime factors**   | \( 3 \cdot 11 \), \( 3 \cdot 17 \), \( 3 \cdot 23 \), \( 11 \cdot 17 \), \( 11 \cdot 23 \), \( 17 \cdot 23 \) |
| **Factors with exactly three prime factors** | \( 3 \cdot 11 \cdot 17 \), \( 3 \cdot 11 \cdot 23 \), \( 3 \cdot 17 \cdot 23 \), \( 11 \cdot 17 \cdot 23 \) |

Would you like further explanations on each of the steps? 

Here are some questions to deepen your understanding:

1. How would the table change if \( N \) had a fifth prime factor?
2. Why do we leave answers in factored form rather than multiplying them out?
3. How can this method be applied to find the factors of a product with non-prime numbers?
4. What role does the concept of combinations play in determining the factors?
5. How many factors would there be if we were to consider factors with all four prime factors?

**Tip:** For any product of distinct primes, the total number of factors is given by \( 2^n - 1 \), where \( n \) is the number of prime factors.

Complete the table if N = 3 * 11 * 17 * 23. Leave all of your answers in factored form.

Learn how to complete a factorization table for N = 3 * 11 * 17 * 23. This exercise covers the concepts of factors, prime factorization, and combinatorics to find factors with exactly two or three prime factors. Ideal for middle school students in Grades 6-8.

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