Difference between revisions of "2011 AIME I Problems/Problem 11"
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Revision as of 19:25, 4 July 2013
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
Note that and . So we must find the first two integers and such that and and . Note that and will be greater than 2 since remainders of will not be possible after 2 (the numbers following will always be congruent to 0 modulo 8). Note that (see Euler's theorem) and are all distinct modulo 125. Thus, and are the first two integers such that . All that is left is to find in mod . After some computation:
See also
2011 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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