Fermat's Little Theorem

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October 19th 2009, 04:07 PM
kyldn6

Fermat's Little Theorem

Prove that $12\ |\ n^2-1$ if g.c.d. (n,6) = 1.
October 19th 2009, 07:32 PM
chisigma

If g.c.d. (n,6) = 1 that means that $2 \nmid n$ and $3 \nmid n$ , i.e. n is and odd number non multiple of 3. But if $2 \nmid n$ and $3 \nmid n$ means also that $n-1$ and $n+1$ are both even numbers and 3 devides one of them. The conclusion is that...

$2\cdot 2\cdot 3 \mid (n+1)\cdot (n-1)$

Kind regards

$\chi$ $\sigma$
October 19th 2009, 09:47 PM
tonio

Quote:

Originally Posted by chisigma

If g.c.d. (n,6) = 1 that means that $2 \nmid n$ and $3 \nmid n$ ,

i.e. n is and odd number non multiple of 3. But if $2 \nmid n$ and $3 \nmid n$ means also that $n-1$ and $n+1$ are both even numbers and 3 devides one of them. The conclusion is that...

$2\cdot 2\cdot 3 \mid (n+1)\cdot (n-1)$

Kind regards

$\chi$ $\sigma$

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