Is it true that $13|2^n5^{2n+1}+3^{2n+1}7^{n+1}$ for all $n$?

Question

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!

Answer 1

Yes it is correct and very nice. The only minor note, we can also conclude at that step of course $$...\equiv 13\cdot 11^n \mod 13$$

As an alternative you can try also by induction.

Answer 2

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*}