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<p>Using binomial expansion of ${{\left( 1+x \right)}^{n}}$, or otherwise, show that</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{7}^{n}}\left( 3n+1 \right)-1$ is always divisible by $9$, for integer $n\ge 1$,</div></li><li><div class="list-item-paragraph">${{9}^{n}}-8n-1$ is divisible by $64$, for integer $n\ge 1$.</div></li></ol>

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

Divisibility Proofs Using Binomial Expansions

Binomial Theorem

<p>Using binomial expansion of ${{\left( 1+x \right)}^{n}}$, or otherwise, show that</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{6}^{n}}-1$ is always divisible by $5$, for integer $n\ge 1$,</div></li><li><div class="list-item-paragraph">${{37}^{11}}-1$ is divisible by $36$.<br><br></div></li></ol>

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