## Task 2 - 140832

Two prime numbers are required to meet the following conditions:

a) Their sum is a prime number.

b) If this sum is multiplied by the product of the two initially mentioned prime numbers, the result is a number divisible by 10.

List all prime numbers that satisfy these conditions.
Assume that two prime numbers $P_{1}, P_{2}$ have the required properties. One of the prime numbers $P_{1}$ and $P_{2}$ must be 2 due to (a), since the sum of odd prime numbers is always greater than 2 and divisible by 2. Without loss of generality, let $P_{1}=2$.

Due to (b), there exists a natural number $n$ such that

$$
\left(2+P_{2}\right) \cdot 2 P_{2}=10 n \quad \text { so } \quad\left(2+P_{2}\right) P_{2}=5 n
$$

is true. Due to the uniqueness of the factorization of a natural number into prime factors, either $2+P_{2}=5$, i.e., $P_{2}=3$ or $P_{2}=5$.

Thus, the prime numbers $(2 ; 3)$ and $(2 ; 5)$ at most satisfy the conditions. Indeed, they have the required properties; their sums $P_{1}+P_{2}$ are 5 and 7, respectively, each of which is a prime number.

The products $\left(P_{1}+P_{2}\right) P_{1} P_{2}$ are 30 and 70, respectively, each of which is divisible by 10.