## Task $5 / 87$

Determine all three-digit (proper) natural numbers $z \in N$ in the decimal system that are represented by exactly $n$ digits 1 in the number system with base $n \in N$.
According to the problem statement,

$$
100 \leq z=\sum_{i=0}^{n} n^{i}=\frac{n^{n}-1}{n-1} \approx n^{n}-11000. \text{ Due to the monotonicity of the function } f(n)=\frac{n^{2}-1}{n-1}, \text{ the only solution is } n=5, z=781.
$$

Indeed, $z=781=625+125+25+5+1=\sum_{i=0}^{5} 5^{i}$.