Problem 3. The ages of the father and his two children (not twins) are powers of the same prime number. One year ago, the age of each of them was a prime number. How old is the father now, and how old is each of his children? (It is known that the father is less than a hundred years old)
Solution. Let $p^{a}, p^{b}, p^{c}$ be the ages of the individuals, where $p$ is a prime number, and $a, b$, and $c$ are natural numbers. According to the problem, one year ago the ages $p^{a}-1, p^{b}-1, p^{c}-1$ were prime numbers. If $p=2$, then the powers of 2 are: $2,4,8,16,32,64,128$, $256, \ldots$. From these, we select those from which when 1 is subtracted, a prime number is obtained. These numbers are 4, 8, and 32 (since $4-1=3, 8-1=7, 32-1=31$). Since the father is less than 100 years old, the only possibility is that the father is 32 years old, one child is 8 years old, and the other is 4 years old.

If $p \geq 3$, then $p^{a}, p^{b}, p^{c}$ are odd numbers. When 1 is subtracted from them, different even numbers are obtained, so all three cannot be prime numbers.