2. Does there exist a natural number, in the representation of which only the digits 7 and 5 appear in equal quantities, and which is divisible by 7 and 5? Justify your answer.
Answer: exists.

Sketch of the solution. Examples are based on the fact that 1001 is divisible by 7. Therefore, a number of the form $\overline{a b c a b c}=\overline{a b c} \times 1001$ is divisible by 7. From this: $777777555555=777777 \times 1000000+555555$.

$5775=5005+770$

For example, the number: 777777555555 or 5775.

Criteria. Correct example, without checking divisibility by 7: 6 points.