Question 37
Let $n$ be a fixed natural number and
$$
S_n = \{\overline{c_n c_{n-1} \dots c_1}_{(10)} \mid c_1, \dots, c_{n-1}, c_n \in \{1, 2, 3, 4\}\}.
$$
Are there distinct numbers $x$ and $y$, $x, y \in S_n$, such that $4^n \mid x - y$?
$$
S_n = \{\overline{c_n c_{n-1} \dots c_1}_{(10)} \mid c_1, \dots, c_{n-1}, c_n \in \{1, 2, 3, 4\}\}.
$$
Are there distinct numbers $x$ and $y$, $x, y \in S_n$, such that $4^n \mid x - y$?