Problem 2. In Grandfather Frost's bag, there are fewer than a hundred gifts for Petya, Vasya, Borya, and Lёsha. Grandfather Frost gave half of the gifts to Petya, a fifth to Vasya, and a seventh to Borya. How many gifts did Lёsha receive?
Answer: 11.

Solution. For Santa Claus to be able to give half of the gifts to Pete, the total number of gifts in his bag must be divisible by 2. Also, since he gave a fifth to Vasya and a seventh to Borya, the total number of gifts must be divisible by 5 and 7. Therefore, the number of gifts must be divisible by $\operatorname{LCM}(2,5,7)=2 \cdot 5 \cdot 7=70$. According to the problem, the number of gifts is less than a hundred, so there can only be 70. Then, he gave Pete $70: 2=35$ gifts, Vasya $-70: 5=14$ gifts, and Borya $-70: 7=10$ gifts. Thus, he gave Lёsha $70-35-14-10=11$ gifts.

## Criteria

1 p. Only the correct answer without justification.

(An answer in the form of a fraction, i.e., $\frac{11}{100}$, is also accepted.)

3 6. Correct reasoning, but there are arithmetic errors.

4 p. The correct answer and justification are provided.

To any of the subsequent points, 1 point is added if the correct answer is given.

1 p. Divisibility of the total number of gifts by 2, 5, or 7 is proven, but there is no further progress.

2 p. It is proven that the number of gifts must be divisible by 70.