[ Equations in integers ] [ GCD and LCM. Mutual simplicity ]

There are fewer than 50 students in the class. For the test, one-seventh of the students received fives, one-third received fours, and half received threes. The rest of the works were rated as unsatisfactory. How many such works were there?
Note: The number of students receiving a particular grade is always an integer.

## Solution

We need to find an integer less than 50 that is simultaneously divisible by $7, 3, 2$. The LCM $(2,3,7)=42$. This means there are 42 students in total; 6 of them received fives; 14 received fours; 21 received threes. Therefore, one student received a two.

## Answer

One student.

## [ GCD and LCM. Mutual simplicity] Problem $\underline{88250}$ Topics: [ $\underset{\text { Equations in integers }}{\text { [ }}]$

The tank was full of water. This water was evenly distributed into three buckets. It turned out that in the first bucket, the water occupied half of its volume, in the second bucket, the water occupied $2 / 3$, and in the third bucket, $3 / 4$ of its volume. The tank and all three buckets hold a whole number of liters. What is the smallest volume of the tank for which this situation is possible?

## Solution

An equal amount of water was poured into each bucket, which is $1 / 3$ of the tank's volume. Thus, the volume of the first bucket is $2 \cdot 1 / 3=2 / 3$ of the tank, the volume of the second bucket is $3 / 2 \cdot 1 / 3=1 / 2$ of the tank, and the volume of the third bucket is $4 / 3 \cdot 1 / 3=4 / 9$ of the tank. All these quantities are whole numbers, so the volume of the tank must be divisible by 3, 2, and 9. $\operatorname{LCM}(2,3,9)=18$. Therefore, the minimum capacity of the tank is 18 liters.

## Answer

18 liters.