Let $n \geq 2$ be a given integer. At any point $(i, j)$ with $i, j \in\mathbb{ Z}$ we write the remainder of $i+j$ modulo $n$. Find all pairs $(a, b)$ of positive integers such that the rectangle with vertices $(0, 0)$, $(a, 0)$, $(a, b)$, $(0, b)$ has the following properties:
[b](i)[/b] the remainders $0, 1, \ldots , n-1$ written at its interior points appear the same number of times;
[b](ii)[/b] the remainders $0, 1, \ldots , n -1$ written at its boundary points appear the same number of times.
1. **Define the problem and notation:**
   - We are given an integer \( n \geq 2 \).
   - We need to find pairs \((a, b)\) of positive integers such that the rectangle with vertices \((0, 0)\), \((a, 0)\), \((a, b)\), \((0, b)\) satisfies two conditions:
     - The remainders \(0, 1, \ldots, n-1\) written at its interior points appear the same number of times.
     - The remainders \(0, 1, \ldots, n-1\) written at its boundary points appear the same number of times.

2. **Interior Points Analysis:**
   - The interior points of the rectangle are those points \((i, j)\) where \(1 \leq i \leq a-1\) and \(1 \leq j \leq b-1\).
   - The total number of interior points is \((a-1)(b-1)\).
   - For the remainders \(0, 1, \ldots, n-1\) to appear the same number of times, \((a-1)(b-1)\) must be divisible by \(n\). Thus, \(n \mid (a-1)(b-1)\).

3. **Boundary Points Analysis:**
   - The boundary points of the rectangle are those points \((i, j)\) where \(i = 0\), \(i = a\), \(j = 0\), or \(j = b\).
   - The total number of boundary points is \(2a + 2b - 4\).
   - For the remainders \(0, 1, \ldots, n-1\) to appear the same number of times, \(2a + 2b - 4\) must be divisible by \(n\). Thus, \(n \mid (2a + 2b - 4)\).

4. **Combining Conditions:**
   - We need both \(n \mid (a-1)(b-1)\) and \(n \mid (2a + 2b - 4)\).
   - Let's analyze these conditions separately:
     - \(n \mid (a-1)(b-1)\) implies that either \(n \mid (a-1)\) or \(n \mid (b-1)\).
     - \(n \mid (2a + 2b - 4)\) can be simplified to \(n \mid 2(a + b - 2)\), which implies \(n \mid (a + b - 2)\).

5. **Necessary and Sufficient Conditions:**
   - For the rectangle to satisfy both conditions, we need:
     - \(n \mid (a-1)\) or \(n \mid (b-1)\)
     - \(n \mid (a+1)\) or \(n \mid (b+1)\)

6. **Conclusion:**
   - The pairs \((a, b)\) of positive integers that satisfy the given conditions are those for which:
     - \(n \mid (a-1)\) or \(n \mid (b-1)\)
     - \(n \mid (a+1)\) or \(n \mid (b+1)\)

The final answer is \( \boxed{ (a, b) } \) such that \(n \mid (a-1)\) or \(n \mid (b-1)\) and \(n \mid (a+1)\) or \(n \mid (b+1)\).