(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove: If n is a composite integer larger than 1 and if no prime number less than [itex] \sqrt{n} [/itex] is a factor of n, then there is an integer m such that [itex] n=m^2 [/itex]

3. The attempt at a solution

Proof: Let n be a positive composite integer larger than 1. If n is composite then there exists an integer y such that y|n where y is larger than 1 but smaller than n . And if y divides n then there exists an integer x such yx=n . Now we take the square root of both sides to obtain

[itex] \sqrt{xy} = \sqrt{n} [/itex]

If there are no prime factors less than the square root of n that are a factor of n then the square root of n is prime. The only way the square root of n could be prime is if x=y .

if x did not equal y then the square root of n would not be an integer. Because the only factors of prime numbers are 1 and them selves. the product of 2 different prime numbers would not be a square.

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# Homework Help: Proof about integers.

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