Mock AIME 4 2006-2007 Problems/Problem 10
Problem
Compute the remainder when
is divided by 1000.
Solution
Let and
be the two complex third-roots of 1. Then let
.
Now, if is a multiple of 3,
. If
is one more than a multiple of 3,
. If
is two more than a multiple of 3,
. Thus
, which is exactly three times our desired expression.
We also have an alternative method for calculating : we know that
, so
. Note that these two numbers are both cube roots of -1, so
.
Thus, the problem is reduced to calculating .
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