## Task A-3.5.

The product of a certain number of mutually distinct natural numbers less than 1000 is not divisible by 250. What is the maximum number of numbers that have been multiplied?
## First solution.

The prime factorization of the number 250 is $250=2 \cdot 5^{3}$.

If we take all numbers that are not divisible by 5 and two numbers that are divisible by 5 but not by 25 (e.g., 5 and 10), we will get a product that is not divisible by 250.

Thus, we can take at least 802 numbers.

1 point

If we take at least 803 numbers, then among them there are at least 3 numbers that are divisible by 5 and at least one that is even, so we get a product that is divisible by 250.