1. Determine all two-digit numbers with the following property: the number and the number written with the same digits, but in reverse order, are both prime.
Solution. For the numbers $\overline{a b}$ and $\overline{b a}$ to be prime, the digits $a$ and $b$ must be odd and different from 5. Otherwise, one of them will be divisible by 2 or 5. Therefore, $a, b \in\{1,3,7,9\}$. By checking, it is found that $91=7 \cdot 13$, so the only such numbers are $\{11,13,17,31,37,71,73\}$.