4. (3 points) On the board, 15 different non-integer numbers were written. For each number $x$ of these fifteen, 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: 4.

Solution:

Let's assume there are no more than 3 resulting numbers. Then, the original numbers can have a maximum of 3 different integer parts and 3 different fractional parts, which means there are no more than $3 \cdot 3=9$ variants of the original numbers.

An example for 4 numbers can be constructed in any way: take any 4 natural numbers, and for each $a$ and $b$ (including identical ones), construct the number $a+\frac{1}{b}$. This results in 16 different original numbers, any of which can be discarded.