8-2. Kolya is older than Tolya, and the age of each of them is a whole number less than 100. If you swap the digits of Kolya's age, you get Tolya's age. Moreover, the difference between the squares of their ages is a square of a whole number. How old is each of them?
Answer: Nikolay is 65 years old, Anatoly is 56 years old.

Solution. Let Kolya's age be $\overline{a b}=10 a+b$, where $a$ and $b$ are digits from 0 to 9. Then Anatoly's age is $\overline{b a}=10 b+a, \mathbf{and} a>b$. We have $(10 a+b)^{2}-(10 b+a)^{2}=9 \cdot 11(a+b)(a-b)$. This difference is a square of an integer, so $a+b$ or $a-b$ must be divisible by 11. Since $a$ and $b$ are digits, it is clear that only the equality $a+b=11$ is possible. Therefore, $a-b=11-2 b$ is a square of an integer. A simple enumeration leads to only one option: $a=6, b=5$. Thus, Nikolay's age is 65 years, and Anatoly's age is 56 years.

Criteria. Only the answer - 0 points. It is shown that the difference of squares has the form $9 \cdot 11(a+$ b) $(a-b)-2$ points. Full solution - 7 points.