Prove that \(5^{3n}+1\) is divisible by \(3^{n+1}\) for all nonnegative integers \(n\).
Write 5 = 3 + 2, and expand using the Binomial Theorem. Then appy the Reduction Lemma to finish.
When n = 2,
\(\displaystyle 5^{6} + 1 = 15625+1=15626,\\ \text{and}\\ 3^{3}=27 \\ \text{and} \\ 15626/27 \approx578.74 \)
No good.
