Question

Tìm tất cả các số tự nhiên n sao cho: \(3n+5⋮n-7\)

Answer 1

Ta có :

\(3n+5⋮n-7\)

Mà \(n-7⋮n-7\)

\(\Leftrightarrow\left\{{}\begin{matrix}3n+5⋮n-7\\3n-21⋮n-7\end{matrix}\right.\)

\(\Leftrightarrow26⋮n-7\)

Vì \(n\in N\Leftrightarrow n-7\in N;n-7\inƯ\left(26\right)\)

\(\Leftrightarrow\left\{{}\begin{matrix}n-7=1\Leftrightarrow n=8\\n-7=26\Leftrightarrow n=33\\n-7=2\Leftrightarrow x=9\\n-7=13\Leftrightarrow n=20\end{matrix}\right.\)

Vậy ....

Answer 2

\(3n+5⋮n-7\)

\(\Rightarrow3n-21+26⋮n-7\)

\(\Rightarrow3\left(n-7\right)+26⋮n-7\)

\(\Rightarrow26⋮n-7\)

\(\Rightarrow n-7\inƯ\left(26\right)\)

\(Ư\left(26\right)=\left\{\pm1;\pm2;\pm13;\pm26\right\}\)

\(\Rightarrow\left[{}\begin{matrix}n-7=1\Rightarrow n=8\\n-7=-1\Rightarrow n=6\\n-7=2\Rightarrow n=9\\n-7=-2\Rightarrow n=5\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}n-7=13\Rightarrow n=20\\n-7=-13\Rightarrow n=-6\\n-7=26\Rightarrow n=33\\n-7=-26\Rightarrow n=-19\end{matrix}\right.\)

Vậy...