Question

Tìm số tự nhiên n để:

a) n + 5 chia hết cho n - 2

b) n² + 3 chia hết cho n + 1

Answer 1

a)Ta có:\(n+5⋮n-2\)

\(\Leftrightarrow n-2+7⋮n-2\)

\(\Leftrightarrow7⋮n-2\)

\(\Leftrightarrow n-2\inƯ\left(7\right)\)

Mà \(n\in N\Rightarrow n-2\ge-2\)

\(\Leftrightarrow n-2\in\left\{-1,1,7\right\}\)

\(\Leftrightarrow n\in\left\{1,3,9\right\}\)

b)\(n^2+3⋮n+1\)

\(\Leftrightarrow n^2+n-n+3⋮n+1\)

\(\Leftrightarrow n\left(n+1\right)-n-1+4⋮n+1\)

\(\Leftrightarrow n\left(n+1\right)-\left(n+1\right)+4⋮n+1\)

\(\Leftrightarrow4⋮n+1\)

\(\Leftrightarrow n+1\inƯ\left(4\right)\)

Mà \(n\in N\Rightarrow n+1\ge1\)

\(\Leftrightarrow n+1\in\left\{1,2,4\right\}\)

\(\Leftrightarrow n\in\left\{0,1,3\right\}\)