Natural numbers `k,l,p` and q are such that if a and b are roots of `x^(2) - kx + l=0`, then `a+1/b` and `b+1/a` are roots of `x^(2) -px + q=0`. What is the sum of all possible values of q?

Answer

To solve the problem, we start with the quadratic equation given:

1. **Identify the roots**: The roots of the equation \(x^2 - kx + l = 0\) are denoted as \(a\) and \(b\). By Vieta's formulas, we know:
- \(a + b = k\)
- \(ab = l\)

2. **Find the new roots**: We need to find the new roots which are \(a + \frac{1}{b}\) and \(b + \frac{1}{a}\).

3. **Calculate the sum of the new roots**:
\[ \text{Sum} = \left(a + \frac{1}{b}\right) + \left(b + \frac{1}{a}\right) = (a + b) + \left(\frac{1}{b} + \frac{1}{a}\right) = k + \left(\frac{a + b}{ab}\right) = k + \frac{k}{l} = k\left(1 + \frac{1}{l}\right) \]

4. **Calculate the product of the new roots**:
\[ \text{Product} = \left(a + \frac{1}{b}\right)\left(b + \frac{1}{a}\right) = ab + a \cdot \frac{1}{a} + b \cdot \frac{1}{b} + \frac{1}{ab} = ab + 1 + 1 + \frac{1}{ab} = ab + 2 + \frac{1}{ab} \]
Substituting \(ab = l\):
\[ \text{Product} = l + 2 + \frac{1}{l} \]

5. **Set up the new quadratic equation**: The new quadratic equation with roots \(a + \frac{1}{b}\) and \(b + \frac{1}{a}\) can be expressed as:
\[ x^2 - px + q = 0 \]
where:
- \(p = k\left(1 + \frac{1}{l}\right)\)
- \(q = l + 2 + \frac{1}{l}\)

6. **Find the minimum value of \(l + \frac{1}{l}\)**: Since \(l\) is a natural number, we can find the minimum value of \(l + \frac{1}{l}\) using AM-GM inequality:
\[ l + \frac{1}{l} \geq 2 \quad \text{(with equality when \(l = 1\))} \]
Thus, the minimum value of \(l\) is \(1\).

7. **Calculate \(q\) for \(l = 1\)**:
\[ q = 1 + 2 + 1 = 4 \]

8. **Check for other values of \(l\)**: For \(l = 2\):
\[ q = 2 + 2 + \frac{1}{2} = 4 + 0.5 = 4.5 \quad \text{(not a natural number)} \]
For \(l = 3\):
\[ q = 3 + 2 + \frac{1}{3} = 5 + \frac{1}{3} = 5.33 \quad \text{(not a natural number)} \]
Continuing this way, we find that \(l\) must be \(1\) to keep \(q\) as a natural number.

9. **Conclusion**: The only possible value of \(q\) is \(4\).

### Final Answer:
The sum of all possible values of \(q\) is \(4\).