Ta có: \(\left(5n+2\right)⋮\left(2n+9\right)\) và \(\left(2n+9\right)⋮\left(2n+9\right)\)
\(\Rightarrow\left\{{}\begin{matrix}2\left(5n+2\right)⋮\left(2n+9\right)\\5\left(2n+9\right)⋮\left(2n+9\right)\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\left(10n+4\right)⋮\left(2n+9\right)\\\left(10n+45\right)⋮\left(2n+9\right)\end{matrix}\right.\)
\(\Rightarrow\left(10n+45\right)-\left(10n+4\right)⋮\left(2n+9\right)\)
\(\Rightarrow41⋮\left(2n+9\right)\)
\(\Rightarrow\left(2n+9\right)\inƯ\left(41\right)\)
\(\Rightarrow\left(2n+9\right)\in\left\{\pm1;\pm41\right\}\)
Ta có bảng sau:
| \(2n+9\) | \(-41\) | \(-1\) | \(1\) | \(41\) |
| \(2n\) | \(-50\) | \(-10\) | \(-8\) | \(32\) |
| \(n\) | \(-25\) | \(-5\) | \(-4\) | \(16\) |
Vậy \(n\in\left\{-25;-5;-4;-16\right\}\) thì \(\left(5n+2\right)⋮\left(2n+9\right).\)
