Question 58
Let $k \in \mathbb{N}$ and $\{a_n\}_{n \ge 0}$ and $\{b_n\}_{n \ge 0}$ be two sequences of positive integers such that $a_i, b_i \in \{1, 2, \dots, 9\}$. For all $n \ge 0$
$$
\overline{a_n \dots a_1 a_0} + k \mid \overline{b_n \dots b_1 b_0} + k.
$$
Prove that there is a number $t$ with $1 \le t \le 9$ and $N \in \mathbb{N}$ such that $b_n = t a_n$ for all $n \ge N$.
(Note that $(\overline{x_n x_{n-1} \dots x_0}) = 10^n \times x_n + \dots + 10 \times x_1 + x_0)$
$$
\overline{a_n \dots a_1 a_0} + k \mid \overline{b_n \dots b_1 b_0} + k.
$$
Prove that there is a number $t$ with $1 \le t \le 9$ and $N \in \mathbb{N}$ such that $b_n = t a_n$ for all $n \ge N$.
(Note that $(\overline{x_n x_{n-1} \dots x_0}) = 10^n \times x_n + \dots + 10 \times x_1 + x_0)$