Challenge Level

Let $n$ be a positive integer.

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

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

- 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.*