## Task $1 / 89$

We are looking for all pairs $\left(p_{1} ; p_{2}\right)$ of (in the decimal system) two-digit prime numbers $p_{1}$ and $p_{2}$ with $p_{1}<p_{2}$, that satisfy the following conditions:

1. Neither the units nor the tens digits of $p_{1}$ and $p_{2}$ are prime numbers.
2. The tens digits of $p_{1}$ and $p_{2}$ are not divisible by 3.
3. The sum $p_{1}+p_{2}$ is neither divisible by 10 nor by 13.
Since $p_{1}$ and $p_{2}$ are prime numbers, the only possible digits for their units place are $1, 3, 7, 9$. According to condition 1, 3 and 7 are excluded. For the tens place, only the numbers $1, 4, 8$ remain after conditions 1 and 2. Therefore, $p_{1}$ and $p_{2}$ can take the values $11, 19, 41, 89$ (49 and 81 are not prime numbers!).

Since $p_{1}<p_{2}$, only the pairs $(11, 19), (11, 41), (11, 89), (41, 89)$ satisfy conditions 1 and 2. Of these, only the pair $(19, 89)$ also satisfies condition 3, making it the only solution to the problem.