Find the largest four-digit number, all digits of which are different and which is divisible by 2, 5, 9, and 11.

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The numbers $2, 5, 9$ and 11 have no common divisors, so if a number is divisible by each of them, it is also divisible by their product. That is, the number we are looking for is divisible by $2 \cdot 5 \cdot 9 \cdot 11=990$. Let's list all four-digit numbers that are divisible by 990: 1980, 2970, 3960, 4950, 5940, 6930, 7920, 8910, 9900. The largest of these is 9900, but it has repeating digits. The largest number with all distinct digits is 8910.

## Answer

8910. Problem 108965 Topics: $\left.\begin{array}{l}\text { Motion problems } \\ {[\text { Arithmetic. Mental calculation, etc. ]] }\end{array}\right]$
Difficulty: $2+$
Grades: 7,8

A motorboat set off upstream at 9 o'clock, and at the moment of its departure, a ball was thrown from the boat into the river. At 9:15, the boat turned and started moving downstream. At what time will the boat catch up with the ball, given that its own speed remained constant?

## Solution

The speed of the river current can be disregarded, as it affects the movement of both the boat and the ball equally. Therefore, the distance between the boat and the ball changes only at the speed of the boat itself. Since the boat was moving away from the ball for 15 minutes, it will catch up with the ball 15 minutes after it turned back.

## Answer

At 9:30.