Let S be the set of 100 consecutive positive integers. Is it possible to rearrange these numbers in a circle so that the product of every two adjacent number is a perfect square?

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anonymous · Answer

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shadowfiend · Answer

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anonymous · Answer

Clearly, the exponent on any prime factor of a product must be even. Therefore, the corresponding prime factors of any product's factors must have the same exponential parity. In S there are at most 25 numbers which share a common prime factor of identical exponential parity. Hence, there exists a product containing a prime factor to an odd power that is therefore non-square.