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In the college textbook we use for my "Algebra and Discrete Mathematics" class, there is a problem, I am unable to solve. With this problem, I would normally use proof by induction instead of a direct proof, but the prompt states that I have to write a direct proof. I don't even know how I should approach this problem.

Unfortunately, our math classes are very fast-paced; not much time, if any, is spent on discussing proof-writing, if it even ever is discussed. Yet, writing proofs is required for the exam.

Prove that for every $n \in \mathbb{N}$ is $n^{3} - n $ always divisible by $3$ – using a direct proof.

Guruprasad
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UselessAnkle48
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    Who says induction is not a form of direct proof? What is your definition of direct proof? And please be more specific about your attempts at this problem. – paw88789 Nov 14 '24 at 10:51
  • See also https://math.stackexchange.com/q/1276003/42969 – despite the title, there are also “non-inductive” proofs. – Martin R Nov 14 '24 at 11:03
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    An indirect proof is a proof where you suppose the negation of what you're trying to prove and reach a contradiction. Proving this by induction wouldn't be an indirect proof. – jjagmath Nov 14 '24 at 11:39
  • Direct Proof in elementary way, since $3/n^3-n $ is given. Take(3 cases)$ n =3k, 3k+1,3k+2$ and plug these to $n^3-n$ then simplify it, then factor 3 – Guruprasad Nov 14 '24 at 13:54
  • We were instructed not to use induction. I am aware that that would constitute a direct proof. – UselessAnkle48 Nov 14 '24 at 21:32

1 Answers1

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Another approach. Every number $n \equiv 0$ or $\pm 1 \pmod 3$. Just substitute each possibility.

Deepak
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