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# Prove that is divisible by for all nonnegative integers .

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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