## Task 1 - 220921

Determine all natural numbers $n$ that satisfy the following conditions (1) and (2):

(1) $n-9$ is a prime number.

(2) $n^{2}-1$ is divisible by 10.
Due to (2), $n^{2}-1$ is even, so $n^{2}$ is odd, so $n$ is odd, so $n-9$ is even, so due to (1) $n-9=2$ and thus $n=11$. Indeed, $11^{2}-1=120$ is also divisible by 10.