## Task 4 - 270924

For any two natural numbers $a, b$, that satisfy the inequalities

$$
3 a-2 b \leq 10 \quad(1) \quad ; \quad 3 a+8 b \leq 25
$$

let $S=a+2 b$.

Determine whether there is a largest number among all the numbers $S$ that can be formed in this way!

If this is the case, find this largest possible value of $S$!
From the second inequality, it follows that $b \leq 3$. We now distinguish based on the value that $b$ takes:

1. Case: $b=3$. Then, according to (2), $a \leq 0$, so $a=0$ and $S=6$.
2. Case: $b=2$. Then it follows from (2) that $a \leq 3$, which also satisfies (1), and $S \leq 3+2 \cdot 2=7$.
3. Case: $b \leq 1$. Then it follows from (1) that $a \leq 4$, so $S \leq 4+2 \cdot 1=6$. Therefore, in each case, $S \leq 7$, which is achieved for the pair $(a, b)=(3,2)$, which satisfies both inequalities.