Divisible Factorisations

Can you show that $n^5-n^3$ is always divisible by $24$?
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Problem

Let $n$ be a positive integer.

  1. Factorise $n^{5}-n^{3},$ and show that it is divisible by $24$.

     
  2. Prove that $2^{2n}-1$ is divisible by $3$.

     
  3. If $n-1$ is divisible by $3$, show that $n^{3}-1$ is divisible by $9$.

 

STEP Mathematics I, 1996, Q3. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.