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QUESTION
Show that the equation $x^2\equiv1$ mod 8 has more than 2
incongruent solutions. Why does this not contradict Lagrange's
theorem?
ANSWER
Each of the residues $\pm1,\ \pm3$ satisfies $x^2\equiv 1$ mod 8,
so this equation has more than 2 incongruent solutions. This does
not contradict Lagrange's theorem, as the theorem refers to prime
moduli only, and 8 is not prime.
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