1983 AIME Problems/Problem 6
Problem
Let equal . Determine the remainder upon dividing by .
Contents
[hide]Solution
Solution 1
First, we try to find a relationship between the numbers we're provided with and . We realize that and both and are greater or less than by .
Expressing the numbers in terms of , we get .
Applying the Binomial Theorem, half of our terms cancel out and we are left with . We realize that all of these terms are divisible by except the final term.
After some quick division, our answer is .
Solution 2
Since (the Euler's totient function), by Euler's Totient Theorem, where . Thus .
See Also
1983 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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