Problem 1533

1. **Define the sequence and the string $s$:**
Let $c$ be a fixed positive integer, and let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers such that $a_n < a_{n+1} < a_n + c$ for every positive integer $n$. Let $s$ denote the infinite string of digits obtained by writing the terms in the sequence consecutively from left to right, starting from the first term. For every positive integer $k$, let $s_k$ denote the number whose decimal representation is identical to the $k$ most left digits of $s$.

2. **Express $m$ in terms of its prime factors:**
Let $m = 2^{\alpha} \cdot 5^{\beta} \cdot v$ where $v$ is a positive integer such that $\gcd(10, v) = 1$.

3. Lemma:
There exists a positive integer $K$ such that $\phi(v) \mid d(K)$ and $v \mid K - 1$, where $d(K)$ is the number of digits of $K$ and $\phi$ is Euler's totient function.

4. Proof of Lemma:
Consider the interval $[10^{m \phi(v) - 1}, 10^{m \phi(v) - 1} + m]$. By the pigeonhole principle, there exists some $T$ in this interval such that $m \mid T - 1$. Then $T \cdot 10^{\phi(v) \cdot \max\{\alpha, \beta\}}$ satisfies the conditions of the lemma.

5. **Constructing the sequence $y_j$:**
Let $y_j$ be the number obtained by concatenating $j$ copies of $K$. We need to show that there exists an $a_n$ in the interval $[y_{2m} \cdot 10^{a_1 + c}, y_{2m} \cdot 10^{a_1 + c + 1})$. Since $a_n$ increases by less than $10^{a_1 + c}$ in each step, we can find such an $a_n$.

6. **Finding the least $a_n$:**
Let $a_{p+1}$ be the least such $a_n$. We have $a_1 < a_{p+1}$, so $p$ is a positive integer. Define $r_0 = \overline{a_1 \ldots a_p}$, $r_1 = \overline{a_1 \ldots a_p y_1}$, ..., $r_{2m} = \overline{a_1 \ldots a_p y_{2m}}$.

7. Proving divisibility:
Notice that $r_{n+1} = r_n \cdot 10^{d(K)} + K \equiv r_n + 1 \pmod{v}$. Therefore, for some $l > 0$, we have $v \mid r_l$. Since $r_l = r_{l-1} \cdot 10^{d(K)} + K$ is divisible by $u$ and $\gcd(u, v) = 1$, it implies $m \mid r_l$, which is a $s_i$.

$\blacksquare$