Difference between revisions of "2022 SSMO Speed Round Problems/Problem 6"

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-==Problem==
-Find the smallest odd prime that does not divide <math>2^{75!} - 1</math>.
-==Solution==
-Let this odd prime be <math>p</math>.
-Note that <math>2^{75!} - 1</math> is divisible by <math>p</math> if
-<cmath>
-^{75!} \equiv 1 \pmod{p}
-</cmath>
-or <math>p - 1 \mid 75!</math>.
-As such, <math>p</math> is the smallest prime of the form <math>2q + 1</math> where
-<math>q > 75</math> is also prime.
-After testing some of the primes above 75, we find that <math>q=68</math> is the smallest prime, meaning the answer is <math>\boxed{167}.</math>

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