2011 USAMO Problems/Problem 4
Problem
Consider the assertion that for each positive integer , the remainder upon dividing by is a power of 4. Either prove the assertion or find (with proof) a counterexample.
Solution
We will show that is a counter-example.
Since , we see that for any integer , . Let be the residue of . Note that since and , necessarily , and thus the remainder in question is . We want to show that is an odd power of for some , and thus not a power of .
Let for some odd prime . Then . Since is co-prime to , we have and thus .
Therefore, for a counter-example, it suffices that be odd. Choosing , we have . Therefore, and thus . Since is not a power of , we are done.
See also
2011 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |