\[{\text{Докажите}}{\text{, что }}{5^{2n + 1}} + {2^{n + 4}} + {2^{n + 1}}{\text{ кратно }}23.\]
\[\begin{array}{l}
{5^{2n + 1}} = 5 \cdot {25^n} \equiv 5 \cdot {2^n} \Rightarrow \hfill \\
{5^{2n + 1}} + {2^{n + 4}} + {2^{n + 1}} \equiv 5 \cdot {2^n} + {2^{n + 4}} + {2^{n + 1}} = {2^n} \cdot \left( {5 + 16 + 2} \right)\mathop \equiv \limits_{23} 0 \hfill \\
\end{array}\]
другие варианты задачи
\[{\text{Докажите}}{\text{, что }}{2^{n + 5}} \cdot {3^{4n}} + {5^{3n + 1}} \vdots 37.\]
\[\begin{array}{l}
{3^{4n}} = {81^n}\mathop \equiv \limits_{37} {7^n} \hfill \\
{5^{3n + 1}} = 5 \cdot {125^n}\mathop \equiv \limits_{37} 5 \cdot {14^n} \hfill \\
{2^{n + 5}} \cdot {3^{4n}} + {5^{3n + 1}} \equiv {2^{n + 5}} \cdot {7^n} + 5 \cdot {14^n} = {14^n} \cdot \left( {32 + 5} \right)\mathop \equiv \limits_{37} 0 \hfill \\
\end{array}\]
\[{\text{Докажите}}{\text{, что 17}} \cdot {\text{2}}{{\text{1}}^{2n + 1}} + 9 \cdot {43^{2n + 1}} \vdots 8.\]
\[\begin{array}{l}
{\text{2}}{{\text{1}}^{2n + 1}} = 21 \cdot {\left( {{{21}^2}} \right)^n} \equiv 21 \cdot {1^n} = 21 \hfill \\
{43^{2n + 1}} = 43 \cdot {\left( {{{43}^2}} \right)^n} \equiv 43 \cdot {1^n} = 43 \hfill \\
{\text{17}} \cdot {\text{2}}{{\text{1}}^{2n + 1}} + 9 \cdot {43^{2n + 1}} \equiv 17 \cdot 21 + 9 \cdot 43 = 744 \vdots 8 \hfill \\
\end{array}\]
