What could be the largest prime divisor of a number in the decimal system of the form $\overline{x y x y x y}$?
The prime factors of the number $\overline{x y x y x y}$ will be deduced from the following product representation:

$$
\overline{x y x y x y}=10101 \cdot(10 x+y)=3 \cdot 7 \cdot 13 \cdot 37 \cdot(10 x+y)
$$

From this, it follows that the largest prime factor of $\overline{x y x y x y}$ is 37, or it comes from the prime factors of $10 x+y$. The largest prime factor of numbers in the form of $10 x+y$ could be the largest two-digit prime number, which is 97. Therefore, the sought largest prime factor is 97, and it belongs to the number 979797.