# 2011 AIME I Problems/Problem 11

## Problem

Let be the set of all possible remainders when a number of the form , a nonnegative integer, is divided by . Let be the sum of the elements in . Find the remainder when is divided by .

## Solution 1

Note that the cycle of remainders of will start after because remainders of will not be possible after (the numbers following will always be congruent to 0 modulo 8). Now we have to find the order. Note that . The order is starting with remainder . All that is left is find in mod after some computation. $$ (Error compiling LaTeX. ! Missing $ inserted.)S=2^0+2^1+2^2+2^3+2^4...+2^102\equiv 2^103-1\equiv 8-1\equiv \boxed{007}\mod 1000$.