2011 USAMO Problems/Problem 4
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.
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.
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