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Manager
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24 Oct 2007, 11:45
This topic is locked. If you want to discuss this question please re-post it in the respective forum.

For every positive even integer n, the funciton 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

1, between 2 and 10,

2, between 10 and 20

3, between 20 and 30

4, between 30 and 40

5, greater than 40
Manager
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24 Oct 2007, 11:48
A. 2 is the smallest even prime in the function, so 2 + 1 would be 3.
Senior Manager
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24 Oct 2007, 11:53
pinal2 wrote:
For every positive even integer n, the funciton 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

1, between 2 and 10,

2, between 10 and 20

3, between 20 and 30

4, between 30 and 40

5, greater than 40

h(100) = 2 x 4 x 6 x 8 x ... x 100
= (2 x 1) x (2 x 2) x (2 x 3) ... x (2 x 50)
= (2^50) x 50!

Now what can be the smallest prime that can divide (2^50) x 50! + 1?
Let's call (2^50) x 50! "First Term"

Let start with number 2: 2 can divide First Term but it cannot divide 1. Out
number 3: 3 can divide First Term but it cannot divide 1. Out
...
number 50: 50 can divide First Term but it cannot divide 1.

So the smallest prime number of h(100) + 1 is clearly greater than 50.

Last edited by devilmirror on 24 Oct 2007, 11:59, edited 1 time in total.
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24 Oct 2007, 11:56
This question was discussed many many times in this forum:

In order to better understand this problem , I suggest to plug in some values.

h(n) where n is positive and even (e.g 2,4,6) and h(n) is the prouduct of all even numbers from 2 to n.

n=10

h(10) = 10*8*6*4*2

so after we understand what this problem wants , we can now solve it !

h(100)+1 = (100*98*96*94....6*4*2) + 1 can be simplified to (2^50)*50!+1

what is the smallest prime factor ?

For any factorial + 1 the smallest factor (apart from 1) is greater then any of the members of the factorial

2! + 1 = 3: smallest factor is 3
3! + 1 = 7: smallest factor is 7
4! + 1 = 25: smallest factor is 5
5! + 1 = 121: smallest factor is 11
6! + 1 = 721: smallest factor is 103

50! + 1 = a number greater then 50

hope this will help.

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24 Oct 2007, 14:04
It helped a lot!! Thanks guys!!
24 Oct 2007, 14:04
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