When 330 is divided by 3, the result is 110, which is not a perfect sqaure

You're correct; 110 is not a perfect square. Let's correct the approach.

### Restating the Problem
- \( \frac{n}{3} \) is a perfect square.
- \( \frac{n}{5} \) is an even number.
- \( \frac{n}{7} \) is a fraction (so \( n \) is **not** divisible by 7).
- \( \frac{n}{11} \) is a whole number (so \( n \) **is** divisible by 11).

### Step-by-Step Solution

1. **Condition 1: \( \frac{n}{3} \) is a perfect square.**
   - This means \( n \) must be of the form \( n = 3k^2 \), where \( k \) is some integer.

2. **Condition 2: \( \frac{n}{5} \) is an even number.**
   - \( n \) must be of the form \( n = 10m \), where \( m \) is an integer.

   Combining conditions 1 and 2: \( n \) must be of the form \( n = 30k^2 \) (since \( n \) must be divisible by both 3 and 10).

3. **Condition 3: \( n \) is not divisible by 7.**
   - \( 30k^2 \) must not be divisible by 7.

4. **Condition 4: \( \frac{n}{11} \) is a whole number.**
   - \( n \) must be divisible by 11. So, \( n = 330k^2 \).

### Conclusion
Given these conditions, \( n = 330k^2 \) for some integer \( k \). However, \( k \) must be such that \( 330k^2 \) is not divisible by 7.

Since \( 330 = 2 \times 3 \times 5 \times 11 \) is not divisible by 7, we look for the smallest \( k^2 \) that meets all the conditions. We test:

- \( k^2 = 1 \) ⇒ \( n = 330 \times 1 = 330 \), but \( \frac{330}{3} = 110 \) is not a perfect square.
- \( k^2 = 4 \) ⇒ \( n = 330 \times 4 = 1320 \) ⇒ \( \frac{1320}{3} = 440 \) (not a perfect square).
- \( k^2 = 9 \) ⇒ \( n = 330 \times 9 = 2970 \) ⇒ \( \frac{2970}{3} = 990 \) (not a perfect square).
- \( k^2 = 16 \) ⇒ \( n = 330 \times 16 = 5280 \) ⇒ \( \frac{5280}{3} = 1760 \) (not a perfect square).
- \( k^2 = 25 \) ⇒ \( n = 330 \times 25 = 8250 \) ⇒ \( \frac{8250}{3} = 2750 \) (not a perfect square).
- \( k^2 = 36 \) ⇒ \( n = 330 \times 36 = 11880 \) ⇒ \( \frac{11880}{3} = 3960 \) (not a perfect square).
- \( k^2 = 49 \) ⇒ \( n = 330 \times 49 = 16170 \) ⇒ \( \frac{16170}{3} = 5390 \) (not a perfect square).
- \( k^2 = 64 \) ⇒ \( n = 330 \times 64 = 21120 \) ⇒ \( \frac{21120}{3} = 7040 \) (not a perfect square).
- \( k^2 = 100 \) ⇒ \( n = 330 \times 100 = 33000 \) ⇒ \( \frac{33000}{3} = 11000 \) (not a perfect square).
- \( k^2 = 121 \) ⇒ \( n = 330 \times 121 = 39930 \) ⇒ \( \frac{39930}{3} = 13310 \) (not a perfect square).
- **\( k^2 = 144 \)** ⇒ **\( n = 330 \times 144 = 47520 \)** ⇒ **\( \frac{47520}{3} = 15840 \)** (which is **even**).

This is the smallest possible \( n \) that satisfies all the conditions.

### Final Answer
The smallest number \( n \) that satisfies all the conditions is **47520**.

Would you like further details or have any questions?

### Related Questions:
1. What is the mathematical significance of a perfect square?
2. How can you systematically check if a number is divisible by multiple factors?
3. Why are the conditions of divisibility crucial in number theory?
4. How does the least common multiple (LCM) help in solving such problems?
5. What are other possible methods for solving this type of problem?

### Tip:
When working with divisibility and perfect squares, try squaring the smallest possible integers and work your way up to find the smallest possible solution.

Explore a mathematical problem involving divisibility and perfect squares. Learn how to find the smallest number satisfying specific conditions. This problem is suitable for advanced mathematics enthusiasts.

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