Question

Tìm x nguyên sao cho biểu thức \(\dfrac{x+1}{x^2+2022}\) là số nguyên

Answer 1

\(\dfrac{x+1}{x^2+2022}\) là số nguyên thì:

\(\left(x+1\right)⋮\left(x^2+2022\right)\\ \Rightarrow\left[\left(x+1\right)\left(x-1\right)\right]⋮\left(x^2+2022\right)\\ \Rightarrow\left(x^2+x-x-1\right)⋮\left(x^2+2022\right)\\ \Rightarrow\left(x^2-1\right)⋮\left(x^2+2022\right)\\ \Rightarrow\left(x^2+2022-2023\right)⋮\left(x^2+2022\right)\)

 \(Mà.\left(x^2+2022\right)⋮\left(x^2+2022\right)\\ \Rightarrow2023⋮\left(x^2+2022\right)\\ \Rightarrow x^2+2022\inƯ\left(2023\right)\\ \Rightarrow x^2+2022\in\left\{-289;-119;-17;-7;-1;-2023;1;7;17;119;289;2023\right\}\)

Ta có: \(x^2+2022\ge0\Rightarrow x^2+2022=2023\Rightarrow x^2=1\Rightarrow x=\pm1\)

Vậy \(x=\pm1\) thì biểu thức trên là số nguyên