Find all positive integers $N$ such that the following holds: There exist pairwise coprime positive integers $a,b,c$ with
$$\frac1a+\frac1b+\frac1c=\frac N{a+b+c}.$$
1. We start with the given equation:
   \[
   \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{N}{a + b + c}
   \]
   where \(a\), \(b\), and \(c\) are pairwise coprime positive integers.

2. Multiply both sides by \(abc(a + b + c)\) to clear the denominators:
   \[
   abc \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) (a + b + c) = N \cdot abc
   \]
   Simplifying the left-hand side:
   \[
   (bc + ac + ab)(a + b + c) = N \cdot abc
   \]

3. Expand the left-hand side:
   \[
   (bc + ac + ab)(a + b + c) = abc + bca + bcb + bcc + aca + acb + acc + aba + abb + abc
   \]
   \[
   = abc(a + b + c) + (ab + ac + bc)(a + b + c)
   \]

4. Since \(a\), \(b\), and \(c\) are pairwise coprime, we can deduce that:
   \[
   abc \mid (a + b + c)(ab + bc + ca)
   \]

5. Consider the divisibility condition \(a \mid (a + b + c)(ab + bc + ca)\). Since \(a\) is coprime with \(b\) and \(c\), it must divide \(b + c\):
   \[
   a \mid b + c
   \]
   Similarly, we have:
   \[
   b \mid c + a \quad \text{and} \quad c \mid a + b
   \]

6. Assume \(a \leq b \leq c\). Since \(a\), \(b\), and \(c\) are pairwise coprime, the smallest values they can take are 1, 2, and 3. We test these values:
   - For \((a, b, c) = (1, 1, 1)\):
     \[
     \frac{1}{1} + \frac{1}{1} + \frac{1}{1} = 3 \quad \text{and} \quad \frac{N}{1 + 1 + 1} = \frac{N}{3}
     \]
     \[
     3 = \frac{N}{3} \implies N = 9
     \]

   - For \((a, b, c) = (1, 1, 2)\):
     \[
     \frac{1}{1} + \frac{1}{1} + \frac{1}{2} = 2.5 \quad \text{and} \quad \frac{N}{1 + 1 + 2} = \frac{N}{4}
     \]
     \[
     2.5 = \frac{N}{4} \implies N = 10
     \]

   - For \((a, b, c) = (1, 2, 3)\):
     \[
     \frac{1}{1} + \frac{1}{2} + \frac{1}{3} = 1 + 0.5 + 0.3333 = 1.8333 \quad \text{and} \quad \frac{N}{1 + 2 + 3} = \frac{N}{6}
     \]
     \[
     1.8333 = \frac{N}{6} \implies N = 11
     \]

7. Therefore, the possible values of \(N\) are 9, 10, and 11.

The final answer is \(\boxed{9, 10, 11}\).