During a math class, the teacher wrote a number on the board. One student said, "The number is divisible by 31." The second student said, "The number is also divisible by 30." A third student claimed the number was divisible by 29, a fourth said it was divisible by 28, and so on, until the thirtieth student said the number was divisible by 2.

After that, the teacher announced that only two of the thirty statements were false, and that these two false statements were consecutive.

Which two statements were incorrect?
According to the teacher - assuming he spoke the truth - out of the thirty numbers mentioned, exactly two are not divisors of the written number, and these two are adjacent. Let's denote the smaller of these two by $a$ $(2 \leq a \leq 30)$. The written number cannot be divisible by $2a$, and since $a$ and $2a$ are not adjacent $(a \geq 2)$, $2a$ cannot be among the listed numbers, so $a \geq 16$. Therefore, the written number is divisible by every number less than 16, and thus it is also divisible by twice, four times, and eight times the odd numbers less than 16.

This means that none of the even numbers greater than 16 can be exceptions, because their half, quarter, or eighth would be an odd number less than 16. Furthermore, one of the two false statements must concern an even number. According to the above, the only possible even exception is 16. Since the written number is divisible by 15, the other false statement must have been about 17.

All of this is possible, for example, if the teacher wrote the product of the prime numbers different from 17 and 8 on the board.