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I'm in a number theory course and my teacher explained Fermat's little theorem, but I'm not sure how to find solutions to the congruence.

Don't necessarily need an answer, a method would be much more helpful! Thank you for all help!

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My first inclination is to observe the squares of the members of $[7]$ and find the reduced residues to see if they equal 1, but I figure there must be a generalized method for large values. – Neurax Apr 11 '13 at 3:48
a method for this sort of situations: Try factoring the polynomial first. If the polynomial is reducible, then find its factorisation as products of irreducible oplynomials; to find the irreducible polynomials, one usually employs some specialized skills like FLT or trial and error, etc... . – awllower Apr 12 '13 at 4:45

Hint $\rm\,\ 7\mid x^2\!-1 = (x\!-\!1)(x\!+\!1)\:\Rightarrow\: 7\mid \ldots\ \ \ or\ \ 7\mid \,\ldots,\:$ since $\,7\,$ is prime.

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(+1)simple and clear – chenbai Apr 11 '13 at 3:57
@Neurax, in general you can show that $x^2\equiv1(\mod p)\Leftrightarrow x=\pm1(\mod p)$. – Easy Apr 11 '13 at 4:00
Very clear (+10086) – Xiaolang Apr 11 '13 at 4:19

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