2. A divisor of a natural number is called proper if it is not equal to the number itself and 1. Find all such natural numbers for which the largest proper divisor differs by 3 (in either direction) from the cube of the smallest proper divisor.
Answer: 10 and 22.

Solution. Let $x$ be the smallest proper divisor of the number $N$. Then its largest proper divisor is $x^{3}+3$ or $x^{3}-3$. Both pairs of numbers, $x$ and $x^{3}+3$, as well as $x$ and $x^{3}-3$, are of different parity, so one of them is even. Both numbers are divisors of $N$, hence $N$ is even. Therefore, the smallest proper divisor of $N$ is 2. Since $x=2$, the largest proper divisor is $2^{3}-3=5$ or $2^{3}+3=11$. Clearly, the largest proper divisor is $\frac{N}{x}$. Therefore, $N=2 \cdot 5=10$ or $N=2 \cdot 11=22$.

Comment. 1 point for each of the two correct answers with verification that they fit. The observation that the largest and smallest divisors are of different parity - 2 points. Missing one of the answers - deduct 3 points. For potentially useful ideas and approaches in the absence of a solution - 2-3 points.