## Task A-2.1.

Determine all natural numbers $k$ for which all solutions of the equation $x^{2}-63 x+k=0$ are prime numbers.
## Solution.

Let $x_{1}$ and $x_{2}$ be the solutions of the given equation. Then we have

$$
x_{1}+x_{2}=63, \quad x_{1} x_{2}=k
$$

Since 63 is an odd number, the only way for the sum of two prime numbers to be 63 is if one of those numbers is 2, and the other must then be 61.

In any case, it follows that $k=x_{1} x_{2}=2 \cdot 61=132$.