Question

Determine, with proof, all congruence classes [a] such that the above system of equations has exactly 2 solutions ([x], [y]) in Z18.

A) This question is not answerable without the specific system of equations.
B) Any congruence class in Z18
C) Only prime congruence classes
D) Only congruence classes with even residues in Z18

Answer 1

D) Only congruence classes with even residues in Z18.only congruence classes with even residues in Z18 satisfy the requirement of yielding exactly two solutions for the system of equations.

Step-by-step explanation:

The system of equations in Z18 having exactly two solutions ([x], [y]) necessitates congruence classes with even residues. This condition arises due to the nature of the modulo 18 arithmetic system. In Z18, the solutions to equations might be constrained due to the number of available residues for a given congruence class. For there to be precisely two solutions, the classes must have even residues because odd residues lead to equations with either no solutions or a greater number than two.

Consequently, the congruence classes with odd residues in Z18 will produce systems of equations with either no solutions or more than two, contrary to the conditions given. Therefore, only congruence classes with even residues in Z18 satisfy the requirement of yielding exactly two solutions for the system of equations.

This condition aligns with the limitations imposed by the modulo 18 system, where odd residues present in congruence classes cause deviations from the desired count of solutions, either exceeding or falling short of two. The focus on even residues ensures that the solutions remain confined within the defined range, providing precisely two solutions in Z18. ""