How many positive integers $N$ in the segment $\left[10, 10^{20} \right]$ are such that if all their digits are increased by $1$ and then multiplied, the result is $N+1$?

[i](F. Bakharev)[/i]
1. **Claim:** For every positive integer \( N = \overline{a_n a_{n-1} \ldots a_0} \), the following inequality holds:
   \[
   N+1 \geq (a_n+1)(a_{n-1}+1) \cdots (a_0+1).
   \]

2. **Proof:** We will use induction on \( n \).

   - **Base Case:** For \( n = 0 \), \( N = a_0 \). Then \( N+1 = a_0 + 1 \), which trivially satisfies the inequality.

   - **Inductive Step:** Assume the inequality holds for any positive integer \( \overline{a_k a_{k-1} \ldots a_0} \). Now consider a positive integer \( N = \overline{a_{k+1} a_{k} \ldots a_0} \). We need to show:
     \[
     N+1 \geq (a_{k+1}+1)(a_k+1) \cdots (a_0+1).
     \]
     Notice that:
     \[
     N+1 = 10^{k+1}a_{k+1} + \overline{a_k a_{k-1} \ldots a_0} + 1.
     \]
     We can rewrite the inequality as:
     \[
     N+1 - \prod_{i=0}^{k+1} (a_i+1) = 10^{k+1}a_{k+1} + \overline{a_k a_{k-1} \ldots a_0} + 1 - (a_{k+1}+1)\prod_{i=0}^k (a_i+1).
     \]
     Simplifying further:
     \[
     N+1 - \prod_{i=0}^{k+1} (a_i+1) = a_{k+1}\left(10^{k+1} - \prod_{i=0}^k (a_i+1)\right) + \left(\overline{a_k a_{k-1} \ldots a_0} + 1 - \prod_{i=0}^k (a_i+1)\right).
     \]
     Since \( 0 \leq a_i \leq 9 \) for every \( 0 \leq i \leq k+1 \), we have \( \prod_{i=0}^k (a_i+1) \leq 10^{k+1} \). By the induction hypothesis, \( \overline{a_k a_{k-1} \ldots a_0} + 1 \geq \prod_{i=0}^k (a_i+1) \). Thus:
     \[
     N+1 \geq \prod_{i=0}^{k+1} (a_i+1).
     \]

3. Now, consider any positive integer \( N = \overline{a_n a_{n-1} \ldots a_0} \) satisfying:
   \[
   N+1 = (a_n+1)(a_{n-1}+1) \cdots (a_0+1).
   \]
   We have:
   \[
   a_n \left(10^n - \prod_{i=0}^{n-1} (a_i+1) \right) + \left(\overline{a_n a_{n-1} \ldots a_0} + 1 - \prod_{i=0}^{n-1} (a_i+1)\right) = 0.
   \]
   Since both terms are non-negative, we must have:
   \[
   10^n = \prod_{i=0}^{n-1} (a_i+1).
   \]
   This only holds if \( a_i = 9 \) for every \( 0 \leq i \leq n-1 \). Thus, \( N \) must be of the form \( x\underbrace{99 \ldots 9}_n \).

4. It is easy to check that \( N = x\underbrace{99\ldots 9}_{n} \) satisfies the problem where \( 1 \leq n \leq 20 \) and \( x \) can be any digit.

The final answer is \( \boxed{N = x\underbrace{99\ldots 9}_{n}} \) where \( 1 \leq n \leq 20 \) and \( x \) can be any digit.