Determine the set of all pairs (a,b) of positive integers for which the set of positive integers can be decomposed into 2 sets A and B so that $a\cdot A=b\cdot B$.
1. **Assume \(a\) and \(b\) are coprime:**
   - Without loss of generality, we can assume that \(a\) and \(b\) are coprime. If they are not, we can divide both by their greatest common divisor (gcd) to make them coprime. This simplification does not affect the generality of the solution.

2. **Consider the placement of 1 in sets \(A\) and \(B\):**
   - Since 1 is a positive integer, it must belong to either set \(A\) or set \(B\). Suppose 1 belongs to set \(A\). Then, \(a \cdot 1 = a\) must be in the set \(a \cdot A\). Similarly, if 1 belongs to set \(B\), then \(b \cdot 1 = b\) must be in the set \(b \cdot B\).

3. **Determine the relationship between \(a\) and \(b\):**
   - Since \(a\) and \(b\) are coprime and one of them must divide the other, we can assume without loss of generality that \(a = 1\). This is because if \(a\) divides \(b\), then \(a\) must be 1 (since they are coprime).

4. **Construct the sets \(A\) and \(B\):**
   - Let \(a = 1\). We need to find a way to partition the set of positive integers into sets \(A\) and \(B\) such that \(1 \cdot A = b \cdot B\). This means that for every integer \(k\) in set \(A\), there must be an integer \(m\) in set \(B\) such that \(k = b \cdot m\).

5. **Define the function \(f(k)\):**
   - For each integer \(k\), define \(f(k)\) as the highest power of \(n\) dividing \(k\). This function helps us determine the placement of \(k\) in sets \(A\) and \(B\).

6. **Partition the integers based on \(f(k)\):**
   - Place \(k\) in set \(A\) if \(f(k)\) is odd.
   - Place \(k\) in set \(B\) if \(f(k)\) is even.
   - This ensures that the sets \(A\) and \(B\) are disjoint and cover all positive integers.

7. **Verify the partition:**
   - By construction, the sets \(A\) and \(B\) satisfy the condition \(1 \cdot A = b \cdot B\). This is because the function \(f(k)\) ensures that each integer \(k\) is placed in the correct set based on its divisibility by \(n\).

8. **Generalize the solution:**
   - The solution can be generalized to pairs \((d, nd)\) and \((nd, d)\) where \(d\) and \(n\) are positive integers. This is because the same reasoning applies when scaling the sets by a common factor \(d\).

The final answer is \( \boxed{ (d, nd) } \) and \((nd, d)\) where \(d\) and \(n\) are positive integers.