Is it true that $13|2^n5^{2n+1}+3^{2n+1}7^{n+1}$ for all $n$? Is it true that $13|2^n5^{2n+1}+3^{2n+1}7^{n+1}$ for all $n$?
So what I did was basically simplify the terms on the right $\mod 13$.
$2^n5^{2n+1}+3^{2n+1}7^{n+1} \mod 13$
$= 5\cdot 2^n\cdot 5^n\cdot 5^n+3\cdot 7\cdot 3^{n}\cdot 3^n7^n \mod 13$
$= 5\cdot 50^n+21\cdot 63^n \mod 13$
$= 5\cdot 50^n+21\cdot 63^n \mod 13$
$= 5\cdot 11^n+8\cdot 11^n \mod 13$
$= 13\cdot 11^n \mod 13$
$= 0\cdot 11^n \mod13$
$=0 \mod13$
Thus, for all $n$, $13$ will divide this term, and the statement is true. Can anyone tell me if what I have done is correct/wrong? Would really appreciate it!

Induction ...
\begin{eqnarray*}
5 \times 50^{n+1}+21 \times 63^{n+1}=50(\color{red}{5 \times 50^{n}+21 \times 63^{n} }) +\color{red}{13}  \times 21\times 63^n.
\end{eqnarray*}
