Task B-1.3. For the transportation of rice, there are bags available in sizes of $40 \mathrm{~kg}$ and $60 \mathrm{~kg}$. How many of each type of bag are needed to transport 500 kg of rice, if the bags must be full? Determine all possibilities.
## Solution.

Let $x$ be the number of bags of $40 \mathrm{~kg}$, and $y$ be the number of bags of $60 \mathrm{~kg}$. We must have $x, y \in \mathbb{N}_{0}$ and

$$
\begin{aligned}
40 x+60 y & =500 \\
2 x+3 y & =25
\end{aligned}
$$

Then $2 x=25-3 y$. For even $y$, $25-3 y$ is an odd number, while $2 x$ is even. Therefore, $y$ must be an odd integer less than 8 (since $25-3 y>0$ must hold). Substitute the odd $y$ values in sequence.

$$
\begin{aligned}
& y=1 \quad \Rightarrow \quad x=11 \\
& y=3 \quad \Rightarrow \quad x=8 \\
& y=5 \quad \Rightarrow \quad x=5 \\
& y=7 \quad \Rightarrow \quad x=2
\end{aligned}
$$

These are all the solutions.

(2 points)