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Prove that \(5^{3n}+1\) is divisible by \(3^{n+1}\) for all nonnegative integers \(n\).

 Oct 8, 2020
 #1
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Write 5 = 3 + 2, and expand using the Binomial Theorem.  Then appy the Reduction Lemma to finish.

 Oct 9, 2020
 #3
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When n = 2,

 

\(\displaystyle 5^{6} + 1 = 15625+1=15626,\\ \text{and}\\ 3^{3}=27 \\ \text{and} \\ 15626/27 \approx578.74 \)

 

No good.

 Oct 10, 2020

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