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# For every positive integer n, the function h(n) is defined

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For every positive integer n, the function h(n) is defined [#permalink]

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15 Sep 2006, 16:14
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For every positive integer n, the function h(n) is defined to be the product of all even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is

a) between 2 and 10
b) between 10 and 20
c) between 20 and 30
d) between 30 and 40
e) greater than 40
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Re: GMAT Prep PS- Function and number properties [#permalink]

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22 Jul 2011, 08:26
uvs_mba wrote:
For every positive integer n, the function h(n) is defined to be the product of all even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is

a) between 2 and 10

b) between 10 and 20

c) between 20 and 30

d) between 30 and 40

e) greater than 40

Let's start by looking at an example for the function h(n):
Consider n = 12. Then h(n) = 2 * 4 * 6 * 8 * 10 * 12. This is already a little tedious to calculate, and obviously h(100) would take at least 20 minutes. Look at this a different way - h(12) = 2(1 * 2 * 3 * 4 * 5 * 6) = 2 * 6!

So, h(100) = 2 * 50!
and h(100) + 1 = 2 * 50! + 1. Obviously dividing h(100) + 1 by any number between 1 and 50 will give you a remainder of 1. Thus it's not divisible by any number less than or equal to fifty, and it's largest prime factor will be larger than 50. Therefore the answer is E.

If you can follow this reasoning, then you can follow one of the most beautiful proofs of all time - Euclid's proof that the set of prime numbers is infinite... definitely worth your time.

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Re: GMAT Prep PS- Function and number properties [#permalink]

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22 Jul 2011, 08:46
uvs_mba wrote:
For every positive integer n, the function h(n) is defined to be the product of all even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is

a) between 2 and 10
b) between 10 and 20
c) between 20 and 30
d) between 30 and 40
e) greater than 40

Remember these rules:

N! is divisible by N! and 0<=x<=n (so by every integer from 0 to n).
Alternatively, (N!+1) is NOT divisible by N! or 0<=x<=n

Now, h(n) = product of all even integers from 2 - 100

(100-2)/2+1=50 So there are 50 even integers between 2-100. Hence our n = 50. Now remember the second rule.

Therefore (E)
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Re: GMAT Prep PS- Function and number properties [#permalink]

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23 Jul 2011, 13:21
Answer is E. I got this question wrong as well!
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Re: GMAT Prep PS- Function and number properties   [#permalink] 23 Jul 2011, 13:21
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# For every positive integer n, the function h(n) is defined

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