Đáp án
$\begin{array}{l}
a)3n - 1 = 3.\left( {n + 3} \right) - 10\\
Do:3\left( {n + 3} \right) \vdots \left( {n + 3} \right)\\
\Rightarrow 10 \vdots \left( {n + 3} \right)\\
\Rightarrow \left( {n + 3} \right) \in \left\{ {5;10} \right\}\left( {do:n \ge 0} \right)\\
\Rightarrow n \in \left\{ {2;7} \right\}\\
b)3n - 1 = 3.\left( {n - 2} \right) + 6 - 1\\
= 3\left( {n - 2} \right) + 5\\
DO:3\left( {n - 2} \right) \vdots \left( {2 - n} \right)\\
\Rightarrow 5 \vdots \left( {2 - n} \right)\\
\Rightarrow \left( {2 - n} \right) \in \left\{ { - 5; - 1;1} \right\}\left( {do:2 - n \le 2} \right)\\
\Rightarrow n \in \left\{ {7;3;1} \right\}\\
c)6{n^2} - 1\\
= 3n.\left( {2n - 3} \right) + 9n - 1\\
Do:3n\left( {2n - 3} \right) \vdots \left( {2n - 3} \right)\\
\Rightarrow 9n - 1 \vdots \left( {2n - 3} \right)\\
\Rightarrow 2\left( {9n - 1} \right) \vdots \left( {2n - 3} \right)\\
\Rightarrow 18n - 2 \vdots \left( {2n - 3} \right)\\
\Rightarrow 9\left( {2n - 3} \right) + 25 \vdots \left( {2n - 3} \right)\\
\Rightarrow \left( {2n - 3} \right) \in \left\{ { - 1;1;5;25} \right\}\left( {do:2n - 3 \ge - 3} \right)\\
\Rightarrow 2n \in \left\{ {2;4;8;28} \right\}\\
\Rightarrow n \in \left\{ {1;2;4;14} \right\}\\
d){n^2} - 7\\
= n\left( {n + 2} \right) - 2n - 4 - 3\\
= \left( {n + 2} \right).n - 2\left( {n + 2} \right) - 3\\
\Rightarrow 3 \vdots \left( {n + 2} \right)\\
\Rightarrow \left( {n + 2} \right) = 3\\
\Rightarrow n = 1
\end{array}$
