Prove of the modular equation $(n^3 - n) (mod\;24) = 0$ with induction

Asked Jan 09 '21 at 21:36

Active Jan 09 '21 at 21:36

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I need to prove this statement with induction:
if n is odd then $n^3 - n $ is divisible by 24

I've proved the base case (when $n = 1$) but I only have a small idea of what is to do in the induction step ($p(n) \Longrightarrow p(n+2)$)

asked Jan 09 '21 at 21:36

Matheus Sousa

1

here is another duplicate, that isn't restricted to even $n$. – lulu Jan 09 '21 at 21:40
Thanks. This helps a lot. – Matheus Sousa Jan 09 '21 at 21:40
2
This is easy, $n^3-n=(n-1)n(n+1)$, therefore:
1. 2|(n-1), 2|(n+1) thus $4|n^3-n=(n-1)n(n+1)$
2. either 3|(n-1), 3|n or 3|(n+1)
– yugikaiba Jan 09 '21 at 21:41

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