MathNet 2025-part-01 — 187 problems (EN–VI) Question 99

Let $S = \{0, 1, 2, \dots, 8\}$. How many functions $f : S \to S$ satisfy the following condition: For any elements $x, y, z \in S$, if $x+y-z$ is divisible by $9$, then $f(x)f(y) - f(f(z))$ is also divisible by $9$?