## Task 5 - 160735

Determine all pairs $(x ; y)$ of natural numbers for which the equation $2 x+3 y=27$ is satisfied!
Assume $(x ; y)$ is a pair of natural numbers that satisfies the equation $2 x+3 y=27$. Then it follows that $3 y=27-2 x$, so in particular, $x$ is a multiple of 3. Further, it follows that

$$
y=9-\frac{2}{3} x
$$

Since $y$ is a natural number, $\frac{2}{3} x \leq 9$, thus $x \leq \frac{27}{2}$; therefore, the only possible values for $x$ are: $x=0, x=3, x=6, x=9$ and $x=12$.

From (1), we get $y=9, y=7, y=5, y=3$ and $y=1$ respectively. Therefore, at most the number pairs $(0 ; 9),(3 ; 7),(6 ; 5),(9 ; 3)$ and $(12 ; 1)$ have the required properties.

These pairs indeed have these properties, as they consist of natural numbers, and it holds that

$$
\begin{gathered}
2 \cdot 0+3 \cdot 9=27 \quad ; \quad 2 \cdot 3+3 \cdot 7=27 \quad ; \quad 2 \cdot 6+3 \cdot 5=27 \\
2 \cdot 9+3 \cdot 3=27 \quad ; \quad 2 \cdot 12+3 \cdot 1=27
\end{gathered}
$$