# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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Power Hungry. Topic: NT/Sequences and Series. Level: Olympiad. February 25th, 2006

Problem: (2005 IMO – #4) Determine all positive integers relatively prime to all the terms of the infinite sequence

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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

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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.

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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 .