Question 108
For a positive integer $m$, let $S(m)$ denote the positive integer whose decimal representation is equal to the octal representation of $m$. For example, $S(64) = 100$, $S(100) = 144$, $S(2023) = 3747$. Two positive integers $x, y$ satisfy the relation
$$
\frac{S(x) + S(1)}{S(x + 1)} = \frac{S(y) + S(2023)}{S(y + 2023)} \lt \frac{8}{10}.
$$
What is the smallest possible value of $y$ if $x \lt 100$?
$$
\frac{S(x) + S(1)}{S(x + 1)} = \frac{S(y) + S(2023)}{S(y + 2023)} \lt \frac{8}{10}.
$$
What is the smallest possible value of $y$ if $x \lt 100$?