Lời giải:
Với $x,y$ nguyên dương .Để \(\frac{x+y-1}{y-1}, \frac{x+y-1}{x-1}\in\mathbb{N}\)
\(\Leftrightarrow \left\{\begin{matrix} x+y-1\vdots y-1\\ x+y-1\vdots x-1\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} x\vdots y-1\\ y\vdots x-1\end{matrix}\right.\)
Suy ra: \(\left\{\begin{matrix} x\geq y-1\\ y\geq x-1\end{matrix}\right.\Rightarrow x+1\geq y\geq x-1\)
Nếu \(y=x+1\Rightarrow x+1\vdots x-1\Leftrightarrow x-1+2\vdots x-1\)
\(\Leftrightarrow 2\vdots x-1\Rightarrow x-1\in\left\{1;2\right\}\) (do \(x-1>0\) )
\(\Rightarrow x\in\left\{2,3\right\}\) \(\Rightarrow y\left\{3,4\right\}\) tương ứng
Nếu \(y=x\Rightarrow x\vdots x-1\)\(\Leftrightarrow x-1+1\vdots x-1\Rightarrow 1\vdots x-1\)
\(\Rightarrow x=2\), kéo theo \(y=2\)
Nếu \(y=x-1\Rightarrow x\vdots y-1\) tương đương với \(x\vdots x-2\Leftrightarrow x-2+2\vdots x-2\Rightarrow 2\vdots x-2\)
Từ đây ta dễ dàng tìm được \(x\in\left\{3,4\right\}\Rightarrow y\in\left\{2,3\right\}\) tương ứng
Vậy:
\((x,y)=(2,3),(3,4),(3,2),(4,3),(2,2)\)
