Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$.

P.S.: This is basically the same problem as http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=548550.
1. **Understanding the Problem:**
   We need to find the largest \( n \) for which there exists a sequence \( (a_0, a_1, \ldots, a_n) \) of non-zero digits such that for each \( k \), \( 1 \le k \le n \), the \( k \)-digit number \( \overline{a_{k-1} a_{k-2} \ldots a_0} \) divides the \( (k+1) \)-digit number \( \overline{a_k a_{k-1} a_{k-2} \ldots a_0} \).

2. **Formulating the Condition:**
   For each \( k \), the \( k \)-digit number \( N_k = \overline{a_{k-1} a_{k-2} \ldots a_0} \) must divide the \( (k+1) \)-digit number \( N_{k+1} = \overline{a_k a_{k-1} a_{k-2} \ldots a_0} \).

   This can be written as:
   \[
   N_{k+1} = a_k \cdot 10^k + N_k
   \]
   and we need \( N_k \mid N_{k+1} \).

3. **Exploring the Sequence:**
   Let's start with \( k = 1 \):
   \[
   N_1 = a_0
   \]
   For \( k = 2 \):
   \[
   N_2 = a_1 \cdot 10 + a_0
   \]
   We need \( a_0 \mid (a_1 \cdot 10 + a_0) \).

4. **Checking for \( n = 4 \):**
   We need to find sequences for \( n = 4 \):
   \[
   N_1 = a_0, \quad N_2 = a_1 \cdot 10 + a_0, \quad N_3 = a_2 \cdot 100 + a_1 \cdot 10 + a_0, \quad N_4 = a_3 \cdot 1000 + a_2 \cdot 100 + a_1 \cdot 10 + a_0
   \]
   We need:
   \[
   a_0 \mid (a_1 \cdot 10 + a_0), \quad N_2 \mid (a_2 \cdot 100 + N_2), \quad N_3 \mid (a_3 \cdot 1000 + N_3)
   \]

5. **Finding Valid Sequences:**
   By trial and error or systematic checking, we find the sequences:
   \[
   \overline{53125}, \overline{91125}, \overline{95625}
   \]
   These sequences satisfy the conditions for \( n = 4 \).

6. **Checking for \( n = 5 \):**
   We need to check if there exists a sequence for \( n = 5 \):
   \[
   N_5 = a_4 \cdot 10000 + a_3 \cdot 1000 + a_2 \cdot 100 + a_1 \cdot 10 + a_0
   \]
   We need:
   \[
   N_4 \mid (a_4 \cdot 10000 + N_4)
   \]
   After checking, we find that no such sequence exists for \( n = 5 \).

Conclusion:
\[
\boxed{4}
\]