268. What three digits should be appended to the number 456 so that the resulting six-digit number is divisible by 504? Find all solutions.
$\triangle$ Let's denote the unknown digits in sequential order as $x, y$, and $z$. Then the six-digit number takes the form $\overline{456 x y z}$.

Since $504=7 \cdot 8 \cdot 9$, it might seem that we need to use the divisibility rules for 8 and 9. However, since the digits $x, y$, and $z$ are adjacent, it is easier to find the number $\overline{x y z}$ directly. To do this, we proceed as follows:

$$
\overline{456 x y z}=456000+\overline{x y z}
$$

We divide 456000 by 504 with a remainder:

$$
456000=504 \cdot 904+384
$$

Then

$$
\overline{456 x y z}=504 \cdot 904+384+\overline{x y z}
$$

The sum $384+\overline{x y z}$ must be divisible by 504.

It can be equal to 504:

$$
384+\overline{x y z}=504
$$

from which

$$
\overline{x y z}=504-384=120
$$

It can also be equal to $504 \cdot 2=1008$:

$$
384+\overline{x y z}=1008
$$

from which

$$
\overline{x y z}=1008-384=624.
$$

Answer: The digits form the numbers 120 or 624.