(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that if [tex] n = m^3 - m [/tex] for some integer [tex]m[/tex], then [tex]n[/tex] is a multiple of 6.

2. Relevant equations

The relevant information is we don't know modular arithmetic yet, and the only methods of proof we have available are direct proof, contradiction, and counterexample.

3. The attempt at a solution

I was thinking of trying to prove it by contradiction, by finding something strange happening in all of the cases where n isn't a multiple of six i.e. for n = 6k + 1 for some integer k on up through n = 6k +5 for some integer k.

For example, in the case where n was one greater than a multiple of 6 we'll have

[tex] m^3 - m - 6k - 1 = 0 [/tex]

I was really hoping to be able to find that the roots of the cubic violated our hypothesis that m be an integer, but I'm having a hard time doing this and I don't want to go nuts on it if it won't be right.

I don't want an outright solution, but a hint would be appreciated.

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# Homework Help: Transition to higher mathematics course, proof

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