For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$:
$$p_1\, p_2\, \cdots\, p_8 
\left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right)
~~=~~ N $$

[i]Proposed by Gerhard Woeginger, Austria[/i]
To solve this problem, we need to determine the largest integer \( N \) for which the equation 
\[ p_1 p_2 \cdots p_8 \left( \frac{x_1}{p_1} + \frac{x_2}{p_2} + \cdots + \frac{x_8}{p_8} \right) = N \]
has no solution in positive integers \( x_1, x_2, \ldots, x_8 \).

1. **Rewrite the equation:**
   \[
   p_1 p_2 \cdots p_8 \left( \frac{x_1}{p_1} + \frac{x_2}{p_2} + \cdots + \frac{x_8}{p_8} \right) = N
   \]
   Simplifying the expression inside the parentheses:
   \[
   \frac{p_1 p_2 \cdots p_8}{p_1} x_1 + \frac{p_1 p_2 \cdots p_8}{p_2} x_2 + \cdots + \frac{p_1 p_2 \cdots p_8}{p_8} x_8 = N
   \]
   \[
   p_2 p_3 \cdots p_8 x_1 + p_1 p_3 \cdots p_8 x_2 + \cdots + p_1 p_2 \cdots p_7 x_8 = N
   \]

2. **Consider the sum of the smallest possible values of \( x_i \):**
   Since \( x_i \) are positive integers, the smallest value each \( x_i \) can take is 1. Therefore, the smallest possible value of the sum is:
   \[
   p_2 p_3 \cdots p_8 + p_1 p_3 \cdots p_8 + \cdots + p_1 p_2 \cdots p_7
   \]

3. **Find the largest integer \( N \) that cannot be expressed as such a sum:**
   We need to find the largest integer \( N \) that cannot be written as the sum of the terms \( p_2 p_3 \cdots p_8 x_1 + p_1 p_3 \cdots p_8 x_2 + \cdots + p_1 p_2 \cdots p_7 x_8 \) where \( x_i \) are positive integers.

4. **Use the Frobenius coin problem (Chicken McNugget theorem):**
   The Frobenius number \( g(a_1, a_2, \ldots, a_n) \) for \( n \) pairwise coprime integers \( a_1, a_2, \ldots, a_n \) is the largest integer that cannot be expressed as a non-negative integer combination of \( a_1, a_2, \ldots, a_n \). For two numbers \( a \) and \( b \), the Frobenius number is \( ab - a - b \).

   For more than two numbers, there is no simple closed-form formula, but the problem can be approached by considering the structure of the sums involved.

5. **Apply the theorem to our problem:**
   Since \( p_1, p_2, \ldots, p_8 \) are distinct primes, they are pairwise coprime. The largest integer \( N \) that cannot be expressed as a sum of the form \( p_2 p_3 \cdots p_8 x_1 + p_1 p_3 \cdots p_8 x_2 + \cdots + p_1 p_2 \cdots p_7 x_8 \) is given by the Frobenius number for these products.

   The largest such \( N \) is:
   \[
   p_1 p_2 \cdots p_8 - \left( p_1 p_2 \cdots p_7 + p_1 p_2 \cdots p_6 p_8 + \cdots + p_2 p_3 \cdots p_8 \right)
   \]

6. **Conclusion:**
   The largest integer \( N \) for which the given equation has no solution in positive integers \( x_1, x_2, \ldots, x_8 \) is:
   \[
   \boxed{p_1 p_2 \cdots p_8 - \left( p_1 p_2 \cdots p_7 + p_1 p_2 \cdots p_6 p_8 + \cdots + p_2 p_3 \cdots p_8 \right)}
   \]