 It follows from this fact that all nonzero remainders modulo formula_11 the square of which isn't congruent to formula_25 can be grouped into pairs formula_26 according to the rule that the product of the members of each pair is congruent to formula_25 modulo formula_11 (since by this fact for every formula_29 we can find such an formula_12, uniquely, and vice versa, and they will differ from each other if formula_31 is not congruent to formula_25).If formula_25 is a quadratic nonresidue, this is simply a regrouping of all formula_34 nonzero residues into formula_35 pairs, hence we conclude that formula_36.If formula_25 is a quadratic residue, exactly two remainders were not among those paired, formula_38 and formula_39 such that formula_40.