Đáp án:
$\begin{array}{l}
1)\\
3n + 7 = 3.n - 6 + 13\\
= 3.\left( {n - 2} \right) + 13\\
Do:3\left( {n - 2} \right) \vdots \left( {n - 2} \right)\\
\Rightarrow 13 \vdots \left( {n - 2} \right)\\
\Rightarrow \left( {n - 2} \right) \in \left\{ { - 13; - 1;1;13} \right\}\\
\Rightarrow n \in \left\{ { - 11;1;3;15} \right\}\\
\text{Vậy}\,n \in \left\{ { - 11;1;3;15} \right\}\\
2)\\
3n + 8 \vdots \left( {2n + 1} \right)\\
\Rightarrow 2.\left( {3n + 8} \right) \vdots \left( {2n + 1} \right)\\
\Rightarrow 6n + 16 \vdots \left( {2n + 1} \right)\\
Do:6n + 16 = 6n + 3 + 13\\
= 3.\left( {2n + 1} \right) + 13\\
Do:3.\left( {2n + 1} \right) \vdots \left( {2n + 1} \right)\\
\Rightarrow 13 \vdots \left( {2n + 1} \right)\\
\Rightarrow \left( {2n + 1} \right) \in \left\{ { - 13; - 1;1;13} \right\}\\
\Rightarrow 2n \in \left\{ { - 14; - 2;0;12} \right\}\\
\Rightarrow n \in \left\{ { - 7; - 1;0;6} \right\}\\
\text{Vậy}\,n \in \left\{ { - 7; - 1;0;6} \right\}\\
3)\\
{n^2} + 3n + 6\\
= n.\left( {n + 3} \right) + 6\\
Do:n.\left( {n + 3} \right) \vdots \left( {n + 3} \right)\\
\Rightarrow 6 \vdots \left( {n + 3} \right)\\
\Rightarrow \left( {n + 3} \right) \in \left\{ { - 6; - 3; - 2; - 1;1;2;3;6} \right\}\\
\Rightarrow n \in \left\{ { - 9; - 6; - 5; - 4; - 2; - 1;0;3} \right\}\\
4)\\
P = 1 + 3 + {3^2} + {3^3} + ... + {3^{101}}\\
= \left( {1 + 3 + {3^2}} \right) + \left( {{3^3} + {3^4} + {3^5}} \right) + ... + \\
\left( {{3^{99}} + {3^{100}} + {3^{101}}} \right)\\
= 13 + {3^3}.\left( {1 + 3 + {3^2}} \right) + ... + {3^{99}}.\left( {1 + 3 + {3^2}} \right)\\
= 13 + {3^3}.13 + ... + {3^{99}}.13\\
= \left( {1 + {3^3} + ... + {3^{99}}} \right).13 \vdots 13
\end{array}$
Vậy P chia hết cho 13
