2. [5 points] Several pairwise distinct natural numbers are written on the board. If the smallest number is multiplied by 30, the sum of the numbers on the board will become 450. If the largest number is multiplied by 14, the sum of the numbers on the board will also become 450. What numbers could have been written on the board
Answer: $13,14,17,29$ or $13,15,16,29$.

Solution. Let there be $n$ numbers $x_{1}30 x_{1} \geqslant$ $30 \cdot(13 \cdot 2)>450$, which is impossible. Therefore, $k=1$, from which $x_{1}=13, x_{n}=29$.

Then $30 x_{1}+x_{n}=30 \cdot 13+29=419$, and the sum of the remaining numbers is $450-419=31$, and the remaining numbers are greater than 13 and less than 29. This cannot be only one number, as it is greater than 29. If there are two numbers, then the different natural numbers with a sum of 31 are 17 and 14 or 16 and 15. There cannot be more than two numbers, as their sum exceeds $13 \cdot 3=39$.