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Hard math problem - completely stuck on it...

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omerkehat
 Post subject: Hard math problem - completely stuck on it...
 Post Posted: Sat Aug 20, 2011 1:48 pm 
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would love it if someone can help with this question:

H(n) is a new formula defined as the product of all even numbers lower than n (inclusive). If p is the smallest prime factor of h(100) + 1 what is p?

1. between 2 - 10
2. between 11 - 20
3. between 21 - 30
4. between 31 - 40
5 greater than 40

i know how to calculate h(100) = 2*4*6*....*98*100, and adding 1 to it makes it an uneven number... but this is where i get stuck.
the correct answer is 5 and i have no idea why (and there is no explanation either..).

appreciate anyone who can assist.
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mithunsam
 Post subject: Re: Hard math problem - completely stuck on it...
 Post Posted: Sat Aug 20, 2011 2:29 pm 
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To solve this, you need to know that "no consecutive numbers share common factors other than 1".

For example, 8 & 9 doesn't share any common factor other than 1
Same is the case for 15 & 16, 21 & 22 etc.

Now H(100) = 2 * 4 * 6 * 8 * ... * 100
= (2 * 1) * (2 * 2) * (2 * 3) * (2 * 4) * ... * (2 * 50)
= 2^50 [1 * 2 * 3 * 4 * 5 * ... * 47 * 48 * 49 * 50]

That means, every prime number till 47 is a factor of H(100). Since H(100) and H(100) + 1 cannot share any common factors other than 1, the greatest prime factor for H(100) + 1 should be greater than 47.

ANSWER E
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omerkehat
 Post subject: Re: Hard math problem - completely stuck on it...
 Post Posted: Sat Aug 20, 2011 3:45 pm 
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thank you!

this looks like a difficult question.. hope i dont encounter something similar in my exam.
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jnelson0612
 Post subject: Re: Hard math problem - completely stuck on it...
 Post Posted: Sun Sep 11, 2011 11:19 pm 
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Nice explanation!

What is the original source of this problem?

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Jamie Nelson
ManhattanGMAT Instructor
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