KöMaL 2025 A916

Let $a \geq 3$ be an integer, and define $f(n) = a^n - 1$ for every positive integer $n$. Denote by $f^{(k)}$ the $k$-iterate of $f$, that is, $f^{(1)}(n) = f(n)$ and $f^{(k+1)}(n)=f(f^{(k)}(n))$ for $k \geq 1$. a) Prove that for any positive integer $K$ there exists a positive integer $M$ such that for every integer $1 \leq k \leq K$, the number $f^{(k)}(M)$ is divisible by $M$ if and only if $k$ is divisible by $2025$. b) Does there exist a positive integer $N$ such that for every positive integer $k$, the number $f^{(k)}(N)$ is divisible by $N$ if and only if $k$ is divisible by $2025$?