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

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SUMMARY

The discussion focuses on proving that the expression n(n² - 1)(n + 2) is divisible by 12 for any integer n. The initial approach involves substituting n with k and expressing the equation as k(k² - 1)(k + 2) = 12m for some integer m. A key suggestion is to factor the term (n² - 1) to facilitate the proof. The conversation highlights the need for clarity on whether induction is necessary, as it has not been covered in the course material.

PREREQUISITES
  • Understanding of basic algebraic factorization
  • Familiarity with integer properties and divisibility rules
  • Knowledge of polynomial expressions
  • Basic concepts of mathematical proofs
NEXT STEPS
  • Study the factorization of polynomials, specifically n² - 1
  • Research divisibility rules for integers, particularly for 12
  • Explore mathematical proof techniques, including direct proof and induction
  • Practice problems involving divisibility and factorization in algebra
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Students in introductory algebra courses, mathematics enthusiasts, and anyone interested in understanding proofs related to divisibility and factorization.

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)...
 
Last edited:
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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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