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To solve this problem, we need to determine how many 4-digit numbers have exactly 9 divisors and that all divisors are from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

### Step-by-Step Solution:
1. **Understanding Divisor Count for Integers:**
   - For a number \( N \) to have exactly 9 divisors, it must be in a form that aligns with divisor rules.
   - If \( N = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k} \) (where \( p_i \) are primes and \( e_i \) are their respective powers), the number of divisors of \( N \) is given by \( (e_1 + 1)(e_2 + 1) \cdots (e_k + 1) \).
   - To get exactly 9 divisors, we need:
     - \( 9 = 9 \times 1 \) (which implies \( N = p^8 \)) or
     - \( 9 = 3 \times 3 \) (which implies \( N = p_1^2 \times p_2^2 \), where \( p_1 \) and \( p_2 \) are distinct primes).

2. **Focus on 4-Digit Numbers:**
   - We need \( N \) to be a 4-digit number (between 1000 and 9999).

3. **Case Analysis:**
   - **Case 1**: \( N = p^8 \):
     - The only 4-digit number of this form would be \( 2^8 = 256 \), which is not a 4-digit number.
   - **Case 2**: \( N = p_1^2 \times p_2^2 \):
     - We need two distinct primes \( p_1 \) and \( p_2 \) such that \( N = p_1^2 \times p_2^2 \) results in a 4-digit number.
     - We must also ensure that all divisors of \( N \) come from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

### Calculation for Case 2:
Let’s examine potential pairs \( (p_1, p_2) \):
   - Possible primes are limited to those within the divisor set, namely: \( \{2, 3, 5, 7\} \).
   - We test pairs:
     - \( N = 2^2 \times 3^2 = 4 \times 9 = 36 \) (not 4-digit)
     - \( N = 2^2 \times 5^2 = 4 \times 25 = 100 \) (not 4-digit)
     - \( N = 2^2 \times 7^2 = 4 \times 49 = 196 \) (not 4-digit)
     - \( N = 3^2 \times 5^2 = 9 \times 25 = 225 \) (not 4-digit)
     - \( N = 3^2 \times 7^2 = 9 \times 49 = 441 \) (not 4-digit)
     - \( N = 5^2 \times 7^2 = 25 \times 49 = 1225 \) (4-digit and valid)

After testing, we find that:
   - The only 4-digit number satisfying these conditions is **1225**.

### Conclusion:
Only one 4-digit number, **1225**, has exactly 9 divisors and meets the given criteria.

Would you like more details on this solution, or do you have any questions?

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Here are five related questions to explore this topic further:

1. What are the rules for determining the number of divisors of any integer?
2. Can a number with 9 divisors ever be a 5-digit number?
3. How do we find all divisors of a given number systematically?
4. What happens if we change the condition to finding numbers with exactly 8 divisors?
5. How would the solution change if we only considered prime numbers up to 20?

**Tip:** The divisor count formula is useful in many contexts, especially when dealing with factors or prime powers in number theory problems.

How many 4-digit numbers have exactly 9 divisors from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}?

Discover how to find the 4-digit numbers that have exactly 9 divisors, specifically from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. This solution involves number theory concepts and the divisor count formula, suitable for students in Grades 10-12.

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