Problem 2. (3 points)

Alice the Fox thought of a two-digit number and told Pinocchio that this number is divisible by $2, 3, 4, 5$, and $6$. However, Pinocchio found out that exactly two of these five statements are actually false. What numbers could Alice the Fox have thought of? In your answer, indicate the number of possible variants.
# Answer: 8

## Solution:

If a number is not divisible by 2, then it is not divisible by 4 or 6 either, and we already have three false statements. Therefore, the number must be divisible by 2. If an even number is divisible by 3, it is also divisible by 6, which means the statements about divisibility by 4 and 5 are false. If the number is not divisible by 3, it is not divisible by 6, and the statements about divisibility by 4 and 5 should be true instead.

In the first case, we get numbers that are divisible by 6 but not by 4 or 5. There are exactly five such two-digit numbers: 18, 42, 54, 66, and 78. In the second case, we get numbers that are divisible by 20 but not by 3. There are three such numbers: 20, 40, and 80. In total, we have 8 numbers.