2. [5 points] Several pairwise distinct natural numbers are written on the board. If the smallest number is multiplied by 32, the sum of the numbers on the board will become 477. If the largest number is multiplied by 14, the sum of the numbers on the board will also become 477. What numbers could have been written on the board?
Answer: $13,14,16,31$ or $13,30,31$.

Solution. Let there be $n$ numbers $x_{1}32 x_{1} \geqslant 32 \cdot(13 \cdot 2)>477$, which is impossible. Therefore, $k=1$, from which $x_{1}=13, x_{n}=31$.

Then $32 x_{1}+x_{n}=32 \cdot 13+31=447$, and the sum of the remaining numbers is $477-447=30$, and the remaining numbers are greater than 13 and less than 31. If there is only one number, it equals 30. If there are two, the only different natural numbers with a sum of 30 are 14 and 16. There cannot be more than two numbers, as their sum would exceed $13 \cdot 3=39$.