## PROBLEM 2

Solve in the set of integers the equation: $5(x-1)+7(y+3)=6(x-1)(y+3)$.

(Mathematical Gazette)
## Solution:

$$
\begin{aligned}
& 5 x - 5 + 7 y + 21 = 6(x y + 3 x - y - 3) \\
& 13 x - 13 y + 6 x y = 34 \\
& 2 \text{p} \\
& x = \frac{34 + 13 y}{6 y + 13} \\
& x \in \mathbb{Z} \Rightarrow \frac{34 + 13 y}{6 y + 13} \in \mathbb{Z} \ldots \cdot 6 \\
& 1 \text{p} \\
& 13 + \frac{45}{6 y + 13} \in \mathbb{Z}, 13 \in \mathbb{Z} \Rightarrow 6 y + 13 \in \text{D}_{45}.
\end{aligned}
$$

$$
\begin{aligned}
& y \in \{-2, -3\} \\
& 1 \text{p} \\
& x \in \{8, -1\} \\
& 1 \text{p}
\end{aligned}
$$