Mock AIME 2 Pre 2005 Problems/Problem 9
Problem
Let where and . Determine the remainder obtained when is divided by .
Solution
We begin by determining the value of . Experimenting, we find the first few s:
We observe that because , will be determined by the base 2 expansion of i. Specifically, every 1 in the s digit of the expansion corresponds to adding to . Since base 2,
Now we look for ways to attain an element with degree . Since each sum of powers of 3 is unique, there is only one; namely, take the x element for every binomial with a degree of one of the added powers of 3 in , and the 1 for all else. Finally, since the coefficients of the x elements are equal to the degree to which the 3 is raised, we conclude
-MRGORILLA
See also
Mock AIME 2 Pre 2005 (Problems, Source) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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