Divisibility Proof for n(n²-1)(n+2) by 12 using Factorization

AI Thread Summary
To prove that n(n² - 1)(n + 2) is divisible by 12 for any integer n, it is suggested to start by factoring n² - 1 into (n - 1)(n + 1). The expression can then be rewritten as n(n - 1)(n + 1)(n + 2). This product includes four consecutive integers, ensuring at least one of them is divisible by 2 and at least one is divisible by 3, thus confirming divisibility by 12. The discussion emphasizes that induction is not necessary for this proof, as the factorization approach suffices. The proof can be completed by demonstrating that the product of these factors meets the criteria for divisibility by 12.
twoski
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Homework Statement



Prove that for any n ∈ Z, n(n² − 1)(n + 2) is divisible by 12 .

The Attempt at a Solution



We first assume n = k for some value k.

Next we assume k(k² − 1)(k + 2) = 12m for some value m.

I don't know where to go from here. I don't think this is supposed to be an induction proof because our professor never explained induction to us yet. Every other proof I've seen for questions like this use induction (because it's so much easier to)...
 
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hi twoski! :smile:

(try using the X2 button just above the Reply box :wink:)

hint: factor (n2 - 1) :wink:
 

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