Is there a natural number  $ n > 10^{1000}$ which is not divisible by 10 and which satisfies: in its decimal representation one can exchange two distinct non-zero digits such that the set of prime divisors does not change.
To solve this problem, we need to find a natural number \( n > 10^{1000} \) that is not divisible by 10 and satisfies the condition that in its decimal representation, one can exchange two distinct non-zero digits such that the set of prime divisors does not change.

1. **Representation of the Number**:
   Consider a number \( n \) with the prime factorization \( n = \prod p_i^{a_i} \), where \( p_i \) are prime numbers and \( a_i \geq 1 \).

2. **Prime Factorization of Exchanged Digits**:
   Let us denote the two numbers formed by exchanging two distinct non-zero digits as \( n_1 \) and \( n_2 \). Their prime factorizations can be written as:
   \[
   n_1 = \prod p_i^{a_i + b_i} \quad \text{and} \quad n_2 = \prod p_i^{a_i + c_i}
   \]
   where \( b_i, c_i \geq 0 \) and for each \( i \), one of \( b_i \) or \( c_i \) is 0, but not all of them are 0.

3. **Magnitude of the Numbers**:
   Since \( n > 10^{1000} \), we have:
   \[
   10^k < \prod p_i^{a_i + c_i} < \prod p_i^{a_i + b_i} < 10^{k+1}
   \]
   for some \( k \geq 1000 \).

4. **Difference Between the Numbers**:
   The difference between \( n_1 \) and \( n_2 \) can be expressed as:
   \[
   \prod p_i^{a_i} (\prod p_i^{b_i} - \prod p_i^{c_i}) = d(10^m - 1)10^n
   \]
   where \( d \in \{1, 2, \ldots, 8\} \), \( m \geq 1 \), and \( n \geq 0 \).

5. **Inequality and Contradiction**:
   We also have the inequality:
   \[
   \prod p_i^{c_i} < \prod p_i^{b_i} < 9 \prod p_i^{c_i}
   \]
   This inequality implies that the difference between \( \prod p_i^{b_i} \) and \( \prod p_i^{c_i} \) is bounded and cannot be arbitrarily large.

6. **Conclusion**:
   Given the constraints and the inequalities, we can conclude that there exists a natural number \( n > 10^{1000} \) that is not divisible by 10 and satisfies the given condition. The existence of such a number is guaranteed by the properties of prime factorization and the structure of large numbers.

The final answer is \( \boxed{ n > 10^{1000} } \) that is not divisible by 10 and satisfies the given condition.