Đề bài suy ra \(x^3+x⋮xy-1\Rightarrow x^3y+xy⋮xy-1\)
\(\Rightarrow x^2\left(xy-1\right)+xy-1+x^2+1⋮xy-1\)
\(\Rightarrow x^2+1⋮xy-1\Rightarrow x^2y+y⋮xy+1\text{}\)
\(\Rightarrow x\left(xy-1\right)+x+y⋮xy-1\Rightarrow x+y⋮xy-1\)
\(\Rightarrow x+y\ge xy-1\)(1)
WLOG \(x\le y \)
+) \(x\ge 3\) thì \(xy-1\ge3y-1>x+y\) (trái với (1)) nên vô lí
(-)\(x=2\Rightarrow y+2⋮2y-1\Rightarrow2y-1+5⋮2y-1\)
\(\Rightarrow5⋮2y-1\Rightarrow2y-1\inƯ\left(5\right)\Rightarrow y=3\left(tm\right)\)
(-)\(x=1\Rightarrow y+1⋮y-1\Rightarrow y+1-2⋮y-1\)
\(\Rightarrow2⋮y-1\Rightarrow y-1\inƯ\left(2\right)\Rightarrow y=2;y=3\left(tm\right)\)
Vậy \(\left(x;y\right)\in\left\{\left(2;3\right),\left(3;2\right),\left(1;2\right),\left(2;1\right),\left(1;3\right),\left(3;1\right)\right\}\)
