Determine all pairs of positive integers $(a,b)$, such that the roots of the equations \[x^2-ax+a+b-3=0,\]
\[x^2-bx+a+b-3=0,\] are also positive integers.
To determine all pairs of positive integers \((a, b)\) such that the roots of the equations 
\[x^2 - ax + a + b - 3 = 0,\]
\[x^2 - bx + a + b - 3 = 0,\]
are also positive integers, we can use Vieta's formulas. 

1. **Using Vieta's Formulas:**
   For the quadratic equation \(x^2 - ax + a + b - 3 = 0\), let the roots be \(k\) and \(l\). By Vieta's formulas, we have:
   \[
   k + l = a \quad \text{and} \quad kl = a + b - 3.
   \]
   Similarly, for the quadratic equation \(x^2 - bx + a + b - 3 = 0\), let the roots be \(m\) and \(n\). By Vieta's formulas, we have:
   \[
   m + n = b \quad \text{and} \quad mn = a + b - 3.
   \]

2. **Equating the Products:**
   Since both equations have the same constant term \(a + b - 3\), we equate the products of the roots:
   \[
   kl = mn = a + b - 3.
   \]

3. **Expressing the Sum of Roots:**
   We also know:
   \[
   k + l = a \quad \text{and} \quad m + n = b.
   \]

4. **Combining the Equations:**
   We can combine these equations to form:
   \[
   \frac{kl + mn}{2} = \frac{a + b - 3 + a + b - 3}{2} = a + b - 3.
   \]
   Simplifying, we get:
   \[
   kl + mn - 2k - 2l - 2m - 2n + 6 = 0.
   \]

5. **Rewriting the Equation:**
   Rearranging the terms, we get:
   \[
   (k-2)(l-2) + (m-2)(n-2) = 2.
   \]

6. **Finding Integer Solutions:**
   We need to find positive integer solutions \((k, l, m, n)\) such that:
   \[
   (k-2)(l-2) + (m-2)(n-2) = 2 \quad \text{and} \quad kl = mn.
   \]

7. **Possible Solutions:**
   Solving the system, we find the following solutions:
   \[
   (k, l, m, n) = (1, 1, 1, 1), (2, 6, 3, 4), (3, 3, 3, 3).
   \]
   These solutions and their permutations give us the pairs \((a, b)\):
   \[
   a = k + l, \quad b = m + n.
   \]

8. **Checking the Solutions:**
   - For \((k, l, m, n) = (1, 1, 1, 1)\):
     \[
     a = 1 + 1 = 2, \quad b = 1 + 1 = 2.
     \]
     The roots are \(1\) and \(1\), which are positive integers.
   - For \((k, l, m, n) = (2, 6, 3, 4)\):
     \[
     a = 2 + 6 = 8, \quad b = 3 + 4 = 7.
     \]
     The roots are \(2, 6\) and \(3, 4\), which are positive integers.
   - For \((k, l, m, n) = (3, 3, 3, 3)\):
     \[
     a = 3 + 3 = 6, \quad b = 3 + 3 = 6.
     \]
     The roots are \(3\) and \(3\), which are positive integers.

9. **Final Solutions:**
   The pairs \((a, b)\) that satisfy the conditions are:
   \[
   (2, 2), (8, 7), (7, 8), (6, 6).
   \]

The final answer is \(\boxed{(2, 2), (8, 7), (7, 8), (6, 6)}\).