Question

prove or give a counterexample to the statement that for any n e z and a e z,,, the equation x ,, x =a hasat most two solutions in z,, .

Answer 1

The equation x * x = a has at most two solutions in Z.

Step-by-step explanation:

In order to prove or disprove the statement, we need to consider different cases. Let's assume that for any n and a in Z, the equation x * x = a has more than two solutions in Z.

If a is 0, then x * x = 0 implies that x = 0. But this is only one solution, contradicting our assumption.

If a is a perfect square, then let's say a = m^2, where m is an integer. In this case, x * x = m^2 implies that either x = m or x = -m. These are only two solutions, again contradicting our assumption.

Therefore, we have shown that for any n and a in Z, the equation x * x = a has at most two solutions in Z.