A positive integer $n$ is called [i]good[/i] if for all positive integers $a$ which can be written as $a=n^2 \sum_{i=1}^n {x_i}^2$ where $x_1, x_2, \ldots ,x_n$ are integers, it is possible to express $a$ as $a=\sum_{i=1}^n {y_i}^2$ where $y_1, y_2, \ldots, y_n$ are integers with none of them is divisible by $n.$ Find all good numbers.
To determine all good numbers \( n \), we need to analyze the given conditions and prove the necessary properties. Let's break down the problem and solution step by step.

1. **Definition and Initial Setup:**
   A positive integer \( n \) is called *good* if for all positive integers \( a \) which can be written as \( a = n^2 \sum_{i=1}^n {x_i}^2 \) where \( x_1, x_2, \ldots, x_n \) are integers, it is possible to express \( a \) as \( a = \sum_{i=1}^n {y_i}^2 \) where \( y_1, y_2, \ldots, y_n \) are integers with none of them divisible by \( n \).

2. **Case Analysis for \( n \):**
   We need to check the conditions for different values of \( n \).

3. **Case \( n = 1 \):**
   - For \( n = 1 \), \( a = 1^2 \sum_{i=1}^1 {x_i}^2 = x_1^2 \).
   - Any \( a \) can be written as \( y_1^2 \) where \( y_1 = x_1 \), and since \( y_1 \) must not be divisible by 1, this condition is trivially satisfied.
   - Therefore, \( n = 1 \) is a good number.

4. **Case \( n = 2 \):**
   - For \( n = 2 \), \( a = 2^2 \sum_{i=1}^2 {x_i}^2 = 4(x_1^2 + x_2^2) \).
   - We need to express \( a \) as \( y_1^2 + y_2^2 \) where \( y_1 \) and \( y_2 \) are not divisible by 2.
   - Consider \( a = 4 \). We have \( 4 = 2^2 + 0^2 \), but 0 is divisible by 2, so this does not satisfy the condition.
   - Therefore, \( n = 2 \) is not a good number.

5. **Case \( n = 4 \):**
   - For \( n = 4 \), \( a = 4^2 \sum_{i=1}^4 {x_i}^2 = 16(x_1^2 + x_2^2 + x_3^2 + x_4^2) \).
   - We need to express \( a \) as \( y_1^2 + y_2^2 + y_3^2 + y_4^2 \) where \( y_1, y_2, y_3, y_4 \) are not divisible by 4.
   - Consider \( a = 64 \). We have \( 64 = 8^2 + 0^2 + 0^2 + 0^2 \), but 0 is divisible by 4, so this does not satisfy the condition.
   - Therefore, \( n = 4 \) is not a good number.

6. **General Case for \( n \geq 3 \) and \( n \neq 4 \):**
   - For \( n \geq 3 \) and \( n \neq 4 \), we need to show that any \( a \) can be written as \( a = \sum_{i=1}^n {y_i}^2 \) where none of the \( y_i \) are divisible by \( n \).
   - If \( n \) does not divide \( 2X \), then \( y_i = 2X - nx_i \) holds the condition.
   - If \( n \) divides \( 2X \):
     - If at least one \( x_i \) is not divisible by \( n \), we can adjust \( X \) to ensure \( y_i \) are not divisible by \( n \).
     - If all \( x_i \) are divisible by \( n \), we can find \( w_i \) such that \( a = \sum_{i=1}^n n^{2k} w_i^2 \) and at least one \( w_i \) is not divisible by \( n \). By applying this iteratively, we can find \( y_i \) not divisible by \( n \).

7. **Conclusion:**
   - From the analysis, we see that \( n = 1 \) is a good number.
   - \( n = 2 \) and \( n = 4 \) are not good numbers.
   - For \( n \geq 3 \) and \( n \neq 4 \), the conditions can be satisfied.

The final answer is \( \boxed{ n = 1 } \) and \( n \geq 3 \) except \( n = 4 \).