2009 AIME I Problems/Problem 8
Revision as of 17:19, 20 March 2009 by Ewcikewqikd (talk | contribs) (New page: == Problem 8 == Let <math>S = \{2^0,2^1,2^2,\ldots,2^{10}\}</math>. Consider all possible positive differences of pairs of elements of <math>S</math>. Let <math>N</math> be the sum of all ...)
Problem 8
Let . Consider all possible positive differences of pairs of elements of . Let be the sum of all of these differences. Find the remainder when is divided by .
Solution
We can do this in an organized way.
If we continue doing this, we will have
Which is
By simplifying this, we will get
We only care about the last three digits So the answer will be