## Task 2 - 140812

Determine all ordered pairs $(x, y)$ of natural numbers $x, y$ for which the equation $13 x+5 y=82$ holds!
(I) Suppose $(x ; y)$ is one of the sought pairs. Then we have

$$
y=\frac{82-13 x}{5} \quad \text { hence } \quad 2 y=\frac{164-26 x}{5}=32-5 x+\frac{4-x}{5}
$$

Since $x$ and $y$ are natural numbers, it follows that $5 \mid 4-x$, i.e., there is an integer $n$ such that $4-x=5 n$, thus $x=4-5 n$. From this, we get

$$
y=\frac{82-13(4-5 n)}{5}=6+13 n
$$

Due to $x \geq 0$, it follows that $4-5 n \geq 0$, hence $n \leq \frac{4}{5}$ and, since $n$ is an integer, thus $n \leq 0$.

Due to $y \geq 0$, it follows that $6+13 n>0$, hence $n \geq-\frac{6}{13}$ and, since $n$ is an integer, thus $n \geq 0$.

Therefore, $n=0$, so $x=4, y=6$. Thus, at most the pair (4;6) can have the required properties.

(II) Indeed, $(4 ; 6)$ is a pair of natural numbers, and it holds that $13 \cdot 4+5 \cdot 6=52+30=82$. The pair $(4 ; 6)$ therefore satisfies the conditions of the problem as the only solution.