Breaking the Hash Function

In some information security problems, so-called hash functions are used. One of the most important classes of hash functions is the so-called polynomial hash functions.

Suppose a string S = s_1 s_2 \ldots s_l consisting of digits from $0$ to $9$ is given. The value of the polynomial hash function p(S, x, m) is computed as follows:

(a \mod b denotes the remainder of the division of $a$ by $b$). For example, let S = 0123, then p(S, 2, 5) = (0 \cdot 1 + 1 \cdot 2 + 2 \cdot 4 + 3 \cdot 8) \mod 5 = 4.

One of the applications of hash functions is storing passwords. Often, passwords have to be stored in an unsecured database table, so instead of storing the passwords themselves, their hash values are stored. When a password is checked, the hash value of the entered string is computed and compared to the value stored in the table.

Your task is: given $x$, $m$, $L$, and $v$, find a string $S$ consisting of digits from $0$ to $9$ of length $L$ such that the value of its polynomial hash function p(S, x, m) is equal to $v$.