Question

Tìm \(x\in Z\) để \(x-2\) chia hết cho \(x^2+1\)

Answer 1

Lời giải:

$x-1\vdots x^2+1(1)$
$\Rightarrow x(x-1)\vdots x^2+1$

$\Leftrightarrow x^2+1-(x+1)\vdots x^2+1$

$\Leftrightarrow x+1\vdots x^2+1(2)$

Lấy $(2)-(1)$ suy ra $2\vdots x^2+1$

$\Rightarrow x^2+1\in\left\{1,2\right\}$ (do $x^2+1>0$)

$\Rightarrow x^2\in\left\{0;1\right\}\Rightarrow x\in\left\{0;\pm 1\right\}$