Question

Tìm số tự nhiên n , biết :

3n + 7 chia hết cho 2n + 1

Answer 1

Vì : \(3n+7⋮2n+1\)

\(\Rightarrow2\left(3n+7\right)⋮2n+1\)

\(\Rightarrow6n+14⋮2n+1\) (1)

Vì : \(2n+1⋮2n+1\)

\(\Rightarrow3\left(2n+1\right)⋮2n+1\)

\(\Rightarrow6n+3⋮2n+1\) (2)

Từ (1) và (2)

\(\Rightarrow\left(6n+14\right)-\left(6n+3\right)⋮2n+1\)

\(\Rightarrow6n+14-6n-3⋮2n+1\)

\(\Rightarrow11⋮2n+1\Rightarrow2n+1\inƯ\left(11\right)\)

\(Ư\left(11\right)=\left\{1;11\right\}\)

+) \(2n+1=1\Rightarrow2n=1-1\Rightarrow2n=0\Rightarrow n=0\)

+) \(2n+1=11\Rightarrow2n=11-1\Rightarrow2n=10\Rightarrow n=5\)

Vậy : \(n\in\left\{0;5\right\}\)