4. (3 points) On the board, 20 different non-integer numbers were written. For each number $x$ from these twenty, Vasya wrote down separately $[x]$ and $\frac{1}{\{x\}}$ in his notebook. What is the smallest number of different numbers Vasya could have obtained?

$[x]$ and $\{x\}$ denote the integer and fractional parts of the number $x$, respectively.
Answer: 5.

Solution:

Let's assume there are no more than 4 numbers in total. Then the original numbers can have a maximum of 4 different integer parts and 4 different fractional parts, which means there are no more than $4 \cdot 4=16$ variants of the original numbers.

An example for 5 numbers can be constructed in any way: for instance, take any 5 natural numbers, and for each different $a$ and $b$, construct the number $a+\frac{1}{b}$.