Re: division

In article <3164991.1115494578013.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
SASWATA SHANNIGRAHI <aj_thak@xxxxxxxxxxx> wrote:
>Am I correct to say that any number (n*n-1)/2 (where n is odd) is not divisible by n?

Is this (1/2)*(n^2 -1) (e.g., for n=3 you would get 4), or is it
(1/2)*n*(n-1) (e.g., for n=3 you would get 3).

If the former, yes, you are correct. If the latter, no, you are
completely wrong.

>Is it an already proved theorem? Otherwise I am ready to give a proof
>which follows directly from Fermat's last theorem.

Why would you do that?

If you mean that n never divides (1/2)*(n^2 - 1), this follows simply
by noting that n^2 -1 is prime to n (since the linear combination n*n
- 1*(n^2-1) is equal to 1), and that n being odd guarantees that
(1/2)(n^2-1) divides n^2 - 1. It is trivial (and known since at least
Euclid) that if a is prime to b, and c divides b, then a is prime to
c.

If you meant that n never divides (1/2)*n*(n-1) for n odd, then I
suggest you try n=1, n=3, n=5 before deducing that in fact the
complete opposite is true: n ALWAYS divides (1/2)*n*(n-1) when n is
odd.


--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

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