2013 Mock AIME I Problems/Problem 7
Problem
Let be the set of all
th primitive roots of unity with imaginary part greater than
. Let
be the set of all
th primitive roots of unity with imaginary part greater than
. (A primitive
th root of unity is a
th root of unity that is not a
th root of unity for any
.)Let
. The absolute value of the real part of
can be expressed in the form
where
and
are relatively prime numbers. Find
.
Solution
.