21.33. If we consider digits in different positions as different, then in a $d$-ary numeral system, $n d$ digits allow us to write $d^{n}$ numbers (from 0 to $d^{n}-1$). Which numeral system is the most economical in this respect, i.e., allows recording the largest number of numbers using a given number of digits? (When comparing numeral systems with bases $d_{1}$ and $d_{2}$, we consider only sets of $m$ digits, where $m$ is divisible by $d_{1}$ and $d_{2}$.)

See also problems $20.15,20.17$.

### 21.10. Other representations of numbers
21.33. Answer: system with base 3. Using $m=d n$ digits, we can write $d^{m / d}$ numbers. Therefore, we need to prove that $3^{m / 3} \geqslant d^{m / d}$, i.e., $3^{d} \geqslant d^{3}$ for any natural $d$. This inequality is proven in the solution to problem 13.11.