February 2017 #0jxq

February 2017Hard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Fermat / Euler / Wilson theoremsQuadratic residuesFactorization techniquesPrime numbers

Problem

Problem: At a recent math contest, Evan was asked to find $2^{2016} \pmod{p}$ for a given prime number $p$ with $100 < p < 500$ . Evan has forgotten what the prime $p$ was, but still remembers how he solved it:

Evan first tried taking $2016$ modulo $p-1$ , but got a value $e$ larger than $100$ .
However, Evan noted that $e - \frac{1}{2}(p-1) = 21$ , and then realized the answer was $-2^{21} \pmod{p}$ . What was the prime $p$ ?

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