## Task 12/90

Determine all triples $(x, y, z)$ of nonnegative integers $x, y$, and $z$ that satisfy the Diophantine equation $3 x+4 y+5 z=30$ and whose sum $s=x+y+z$ is a prime number, through logical reasoning (thus, a solution approach based on systematic trial and error is excluded!).
If $(x, y, z)$ with $x ; y ; z \geq 0$ and integers is a solution triplet of the given Diophantine equation, then for the sum $s=x+y+z$ it holds that $3 s \leq 3 x+4 y+5 z=30$ and $5 s \geq 3 x+4 y+5 z=30$, which gives the inequality $6 \leq s \leq 10$.

Due to the prime number property of $s$, it follows that $s=7=x+y+z$. Subtracting $3 s$ or $4 s$ from the given equation, we get $y=9-2 z, x=z-2$.

Since $x, y \geq 0$, it follows that $2 \leq z \leq 4$. This yields the three solution triplets $(0 ; 5 ; 2),(1 ; 3 ; 3)$ and $(2 ; 1 ; 4)$. Verification confirms that all three triplets satisfy the conditions of the problem.