For a natural number to bring Libor luck, its square must be divisible by the numbers seven, eight, nine, and ten. Find all natural numbers less than 1000 that bring Libor luck.
Let's decompose the given divisors into prime factors:

$$
7 ; \quad 8=2 \cdot 2 \cdot 2 ; \quad 9=3 \cdot 3 ; \quad 10=2 \cdot 5
$$

If the square of a "lucky number" is to have all these divisors, its prime factorization must include this product:

$$
2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 \cdot 7
$$

The "lucky number" as the base of the square must therefore include this product in its prime factorization:

$$
2 \cdot 2 \cdot 3 \cdot 5 \cdot 7=420 .
$$

The "lucky number" is thus a natural multiple of 420. There are only two such numbers less than 1000, namely 420 and 840.

$[2+2$ points for the results, 2 points for explaining the process]