For which integers $n\ge 2$ can the numbers $1$ to $16$ be written each in one square of a squared $4\times 4$ paper such that the $8$ sums of the numbers in rows and columns are all different and divisible by $n$?
1. **Define the sums of rows and columns:**
   For \( k \in \{1,2,3,4\} = S \), let \( R_k \) and \( C_k \) be the sum of the four numbers in row \( k \) and column \( k \) respectively. According to the given information, there exist 8 distinct positive integers \( x_1, x_2, \ldots, x_8 \) such that \( R_k = nx_k \) and \( C_k = nx_{k+4} \) for \( k \in S \).

2. **Determine the maximum possible sums:**
   The maximum sum of any row or column is:
   \[
   \max\{R_k, C_k\}_{k \in S} \leq 16 + 15 + 14 + 13 = 58
   \]
   Since there are 8 distinct sums, the maximum value of \( x_i \) must be at least 8:
   \[
   \max\{x_i\}_{i=1}^8 \geq 8
   \]
   This implies:
   \[
   8n \leq 58 \implies n \leq 7
   \]

3. **Sum of all elements and divisibility condition:**
   The sum of all elements in the \( 4 \times 4 \) grid is:
   \[
   \sum_{m=1}^{16} m = \frac{16 \cdot 17}{2} = 136
   \]
   Since the sums of rows and columns are all distinct and divisible by \( n \), we have:
   \[
   n \sum_{i=1}^8 x_i = \sum_{i=1}^8 nx_i = \sum_{k=1}^4 R_k + \sum_{k=1}^4 C_k = 2 \sum_{m=1}^{16} m = 2 \cdot 136 = 272
   \]
   This implies:
   \[
   n \sum_{i=1}^8 x_i = 272
   \]
   Since \( \sum_{i=1}^8 x_i \) must be an integer, \( n \) must divide 272. The prime factorization of 272 is:
   \[
   272 = 2^4 \cdot 17
   \]
   Therefore, \( n \) must be a divisor of \( 2^4 \cdot 17 \) and within the range \( 2 \leq n \leq 7 \). The possible values of \( n \) are 2 and 4.

4. **Verification with a specific \( 4 \times 4 \) grid:**
   Consider the following \( 4 \times 4 \) grid containing the numbers 1-16:
   \[
   \begin{array}{cccc}
   1 & 3 & 5 & 7 \\
   2 & 4 & 6 & 8 \\
   9 & 11 & 13 & 15 \\
   16 & 14 & 12 & 10 \\
   \end{array}
   \]
   The sums of the rows and columns are:
   \[
   R_1 = 16, \quad R_2 = 20, \quad R_3 = 48, \quad R_4 = 52
   \]
   \[
   C_1 = 28, \quad C_2 = 32, \quad C_3 = 36, \quad C_4 = 40
   \]
   These sums are all distinct and divisible by both \( n = 2 \) and \( n = 4 \).

Conclusion:
The only positive integers \( n \geq 2 \) which satisfy the given conditions are \( n = 2 \) and \( n = 4 \).

The final answer is \( \boxed{ n = 2 } \) and \( n = 4 \).