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Math Help - Function and prime factors

  1. #1
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    Function and prime factors

    Hi! In studying for the GMAT, I ran into a rather difficult problem. I don't know where to start or any of the steps to choosing the correct answer.

    For every positive even integer, N, the function h(n) is defined to be the product of all the even integers from 2 to N inclusive. If P is the smallest prime factor of h(100)+1, then p is:

    Between 2 and 10
    Between 10 and 20
    Between 20 and 30
    Between 30 and 40
    Greater than 40

    The correct answer is Greater than 40, but I am at a loss as to how to find it. Any help would be greatly appreciated.
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  2. #2
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    Re: Function and prime factors

    Note that h(100) = 2^{50}\cdot50!. The proof that h(100) + 1 is not divisible by any number <= 40 is similar to the Euclid's proof that there are infinitely many primes.
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  3. #3
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    Re: Function and prime factors

    Quote Originally Posted by emakarov View Post
    Note that h(100) = 2^{50}\cdot50!. The proof that h(100) + 1 is not divisible by any number <= 40 is similar to the Euclid's proof that there are infinitely many primes.
    Ah I see, thank you very much for your explanation.
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