Problem: (2005 IMO – #4) Determine all positive integers relatively prime to all the terms of the infinite sequence
Solution: We claim that the only such integer is . Consider any prime and the term . We have
which, by taking a common denominator and factoring, becomes
But by Fermat’s Little Theorem, we have , so
So we have shown that for all primes . Noting that , which is divisible by both and , we have that every positive integer divisible by a prime is not relatively prime to all terms in the sequence (since at least one term is divisible by every prime). Hence the only possible number is , as claimed. QED.
Comment: The quickest way to find this solution was seeing that . And since multiplicative inverses always exist modulo a prime, the term is divisible by .
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