Problem:
Let be the number of complex numbers with the properties that and is a real number. Find the remainder when is divided by .
Solution:
If satisfies the given conditions, there is a such that and is real. This difference is real if and only if either the two numbers and represent the same angle or the two numbers represent supplementary angles. In the first case there is an integer such that , which implies that is a multiple of . In the second case there is an integer such that , which implies that is plus a multiple of . In the interval there are values of that are multiples of , there are values that are plus a multiple of , and there are no values of that satisfy both of these conditions. Therefore there must be complex numbers satisfying the given conditions. The requested remainder is .
The problems on this page are the property of the MAA's American Mathematics Competitions