If we write a three-digit number twice side by side, the resulting number is divisible by 7, 11, and 13. If we write a four-digit number twice side by side, we get a number that is divisible by 73 and 137.
1. If the three-digit number is $a$, then the new number is

$$
1000 a+a=1001 a=7 \cdot 11 \cdot 13 a .
$$

2. If the four-digit number is $b$, then the new number is:

$$
10000 b+b=10001 b=73 \cdot 137 b \text {. }
$$

(István Braun, Eger.)

Number of solutions: 54.