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. Unknown error_msg)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$.