1. (10 points) In a $3 \times 5$ table, all numbers from 1 to 15 were arranged such that the sum of the numbers in any column of three cells is divisible by 3. Then some of them were erased and replaced with numbers $a, b, c$:

| 11 | $a$ | 3 | 8 | 10 |
| :---: | :---: | :---: | :---: | :---: |
| 7 | 9 | 13 | 15 | 12 |
| $c$ | 2 | 5 | 4 | $b$ |

Which of the following statements is true? a) $a=1, b=14$; b) $b=14, c=12$; c) $a=4$, $c=6$; d) $a=14, b=6$; e) none of the options are correct.
Answer: $a=1, b=14$.

Solution: For the sum of the numbers in each column to be divisible by three, it is necessary that 1) $c$ is divisible by three; 2) $a$ gives a remainder of $1$ when divided by three; 3) $b$ gives a remainder of $2$ when divided by three. Therefore, the fourth option does not fit.

According to the condition, each number in the table appears exactly once, so options b) and c) are not correct, and the first option ( $a=1, b=14$ ) satisfies this condition. Thus, we get the final answer.