SE-SKOLORNAS 1994 Problem 6

Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$ . Also $f(10) = 1$ . How many three digit numbers $n$ satisfy $f(n) = f(N)$ , where $N$ is the "tower" $2, 3, 4, 5$ , in other words, it is $2^a$ , where $a = 3^b$ , where $b = 4^5$ ?