Given x2≡182 mod 727, how many solutions mod 727 does it have? Note 727 is prime and 182=2⋅7⋅13.
So I know this is soluble computing (182727), but how do I determine the number of solutions? I know how to do this for linear congruences but not sure how to do it for quadratic.
Answer
727≡3mod4 so the computation of ±182727+14=±182182
in modulo 727 will give you the desired solutions. Incidentally, the set of solutions are {363,364}.
As another user alludes in his comment; since Z727 is a field; the equation x2≡amod727 will have at most two solutions.
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