Question

[KK họ Phạm]

Chứng minh rằng:

a,n+5 chia hết cho n+2

b,2n+1 chia hết cho n-5

c,\(n^2\)+3n-13 chia hết cho n+3

d,\(n^2\)+3 chia hết cho n-1

Answer 1

[Nguyễn Nam]

Đề bài là tìm n chứ:

a) Ta có:

\(n+5⋮n+2\)

\(\Rightarrow\left(n+2\right)+3⋮n+2\)

\(\Rightarrow3⋮n+2\)

\(\Rightarrow n+2\in U\left(3\right)=\left\{-1;1;-3;3\right\}\)

\(\Rightarrow\left\{{}\begin{matrix}n+2=-1\Rightarrow n=-3\\n+2=1\Rightarrow n=-1\\n+2=-3\Rightarrow n=-5\\n+2=3\Rightarrow n=1\end{matrix}\right.\)

Vậy \(n\in\left\{-3;-1;-5;1\right\}\)

b) Ta có:

\(2n+1⋮n-5\)

\(\Rightarrow\left(2n-10\right)+11⋮n-5\)

\(\Rightarrow2\left(n-5\right)+11⋮n-5\)

\(\Rightarrow11⋮n-5\)

\(\Rightarrow n-5\in U\left(11\right)=\left\{-1;1;-11;11\right\}\)

\(\Rightarrow\left\{{}\begin{matrix}n-5=-1\Rightarrow n=4\\n-5=1\Rightarrow n=6\\n-5=-11\Rightarrow n=-6\\n-5=11\Rightarrow n=16\end{matrix}\right.\)

Vậy \(n\in\left\{4;6;-6;16\right\}\)

c) Ta có:

\(n^2+3n-13⋮n+3\)

\(\Rightarrow n\left(n+3\right)-13⋮n+3\)

\(\Rightarrow-13⋮n+3\)

\(\Rightarrow n+3\in U\left(13\right)=\left\{-1;1;-13;13\right\}\)

\(\Rightarrow\left\{{}\begin{matrix}n+3=-1\Rightarrow n=-4\\n+3=1\Rightarrow n=-2\\n+3=-13\Rightarrow n=-16\\n+3=13\Rightarrow n=10\end{matrix}\right.\)

Vậy \(n\in\left\{-4;-2;-16;10\right\}\)