Using binomial expansion of ${{\left( 1+x \right)}^{n}}$, or otherwise, show that ${{6}^{n}}-1$

Question

Using binomial expansion of ${{\left( 1+x \right)}^{n}}$, or otherwise, show that ${{6}^{n}}-1$ is always divisible by $5$, for integer $n\ge 1$, ${{37}^{11}}-1$ is divisible by $36$.

Accepted Answer

(i) $5\mid(6^n-1)$ for every integer $n\ge 1$ (ii) $36\mid(37^{11}-1)$

Binomial Theorem - Divisibility Proofs Using Binomial Expansions

Binomial Theorem - Divisibility Proofs Using Binomial Expansions

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