Problem 5. Determine all natural numbers $n$ such that the numbers $3 n-4, 4 n-5$, and $5 n-3$ are prime numbers.
Solution. The sum of the numbers $3 n-4, 4 n-5$, and $5 n-3$ is an even number, so one of them must be an even number. The number $4 n-5$ cannot be even because $4 n$ is even and 5 is odd. Therefore, the numbers $3 n-4$ and $5 n-3$ can only be even. Since 2 is the only even prime number, it can be $3 n-4=2$ or $5 n-3=2$. From this, we get that $n=2$ or $n=1$. However, $n=1$ is not possible. For $n=2$, we get the prime numbers $3 n-4=2, 4 n-5=3$, and $5 n-3=7$. Therefore, $n=2$ is the desired natural number.

## VIII Section