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

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Intern
Joined: 08 Apr 2008
Posts: 26

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21 Apr 2008, 09:05
n is a integer, n>0 and h(n)= the product of all the even integers from 2 to n inclusive. If p is the smallest prime factor of
h(n)+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

OA to follow
pls help!
Intern
Joined: 21 Jun 2007
Posts: 11

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21 Apr 2008, 11:04
Well, I can definitely get get a couple answers that coincide with choice A.

If n = 4, then h(n) = 2x4 = 8

8 + 1 = 9

Smallest (and only) prime factor = 3

If n = 6, then h(n) = 2x4x6 = 48

48 + 1 = 49

Smallest (and only) prime factor = 7

So, I'd go with A

However, the question looks very familiar, are you sure you wrote it correctly?
Intern
Joined: 02 Apr 2008
Posts: 37

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21 Apr 2008, 11:38
Brindusa wrote:
n is a integer, n>0 and h(n)= the product of all the even integers from 2 to n inclusive. If p is the smallest prime factor of
h(n)+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

OA to follow
pls help!

Must be 100, not n, I guess.
7-p446677
Intern
Joined: 28 Mar 2008
Posts: 34

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21 Apr 2008, 14:14
A
if we consider n =1,4 and 6

the close factors for the function I could get are1, 3 and 7
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Joined: 04 May 2006
Posts: 1894
Schools: CBS, Kellogg
Re: GMAT Prep Question on Integers [#permalink]

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25 May 2008, 18:38
Nice reasoning! thanks!
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Posts: 128
Location: Chicago
Re: GMAT Prep Question on Integers [#permalink]

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28 May 2008, 14:05
TheGMATDoctor wrote:
This question is testing the concept of coprimes.
2 positive integers are coprime when their greatest common factor (their only common factor) is 1.
Now note that two different prime numbers are always coprime.
For example, 3 and 7 are coprime. So are 13 and 19.
But the two integers need not be prime numbers in order to be coprime.
For example, 4 and 9 are coprime (1 is their only common factor).
Also, Important!
Two consecutive integers are always coprime. The question is testing you on this concept.
Let's solve it now:
h(n) = 2*4*6*....................*n. ----n is even.
h(100) = 2*4*6*........ 94*96*98*100.
h(100) = (2^50)*(1*2*3*.......47*48*49*50). Note:I have pooled together all the 2s from all the even integers from 2 to 100; that's how I got 2^50. Now, the largest prime number involved in the above factorization is 47. All the prime from 2 to 47 are also involved in the above factorization. Actually, 47 is the greatest prime factor of h(100).
Since h(100) and h(100) + 1 are consecutive integers, they are necessarily coprime (see above). h(100) and h(100) + 1 have no common factor except 1, so they have no common prime factor either. The smallest prime factor of h(100) +1 must then be greater than 47.
Clearly, this prime factor is greater than 40.

That's all folks!
Asan Azu, The GMAT Doctor.

Wow, lot of info. I understood everything except for this part: h(100) = (2^50)*(1*2*3*.......47*48*49*50)
Could you explain this part a little more clearly? Thanks.
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Factorials were someone's attempt to make math look exciting!!!

Manager
Joined: 01 May 2008
Posts: 110
Location: São Paulo

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08 Jun 2008, 12:21
For every positive 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, the 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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Joined: 28 Dec 2004
Posts: 3357
Location: New York City
Schools: Wharton'11 HBS'12

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08 Jun 2008, 12:35
ldpedroso wrote:
For every positive 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, the 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

before posting a question..please search for it on this forum...this question has been solved 1000 times here..
Manager
Joined: 20 Sep 2007
Posts: 106
Function h(n) Gmat prep question [#permalink]

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07 Jul 2008, 21:04
For every positive 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 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

OA TO FOLLOW
Manager
Joined: 05 Jul 2008
Posts: 139
GMAT 2: 740 Q51 V38
Re: Function h(n) Gmat prep question [#permalink]

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07 Jul 2008, 21:14
neeraj.kaushal wrote:
For every positive 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 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

OA TO FOLLOW

My choice is E.
h(100)= 2*4*6....*100 so it contains all the prime number from 2 to 49 so h(100)+1 is not divisible by 2, 3,..., 49 so its smallest prime factor must be greater than 49.
Is it right?
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Joined: 27 May 2008
Posts: 541
Re: Function h(n) Gmat prep question [#permalink]

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07 Jul 2008, 21:20
refer 7-t64451
SVP
Joined: 21 Jul 2006
Posts: 1510
Re: Function h(n) Gmat prep question [#permalink]

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12 Jul 2008, 08:09
There is something I don't understand here. Just because the largest prime factor of h(100) is 47, shouldn't mean that the smallest prime factor of h(100)+1 will be bigger than 47. Let me give you an example:

9 = 3*3, the largest prime number here is 3, so when we look at the next higher number:

10=2*5, so the smallest prime factor here is 2, which is not bigger than 3. So how can we say that the bigger number's smallest prime factor will be bigger than the biggest prime factor of the number just under it??? Can someone please explain this concept to me? thanks
Manager
Joined: 05 Jul 2008
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GMAT 2: 740 Q51 V38
Re: Function h(n) Gmat prep question [#permalink]

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12 Jul 2008, 21:30
tarek99 wrote:
There is something I don't understand here. Just because the largest prime factor of h(100) is 47, shouldn't mean that the smallest prime factor of h(100)+1 will be bigger than 47. Let me give you an example:

9 = 3*3, the largest prime number here is 3, so when we look at the next higher number:

10=2*5, so the smallest prime factor here is 2, which is not bigger than 3. So how can we say that the bigger number's smallest prime factor will be bigger than the biggest prime factor of the number just under it??? Can someone please explain this concept to me? thanks

False analogy
h(100) is divisible by all prime numbers from 2, ..., 47 while 9 is not divisible by all prime numbers from 2 to 3. If 9 were, 10 would never divisible by 2.
If A is divisible by p and A+1 is divisible by p too, 1 is divisible by p. It's wrong.
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Re: Function h(n) Gmat prep question [#permalink]

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13 Jul 2008, 11:48
Manager
Joined: 23 Jun 2008
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GMAT Prep: Function - part 2 [#permalink]

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23 Jul 2008, 17:47
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

OA is e.
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Joined: 27 May 2008
Posts: 141
Re: GMAT Prep: Function - part 2 [#permalink]

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23 Jul 2008, 20:24
h (100) would mean 2*3*4*5*6...............................100 = X
Now X has the following properties:

1) it's an even number.
2) it ends in more than three zeros thus a multiple of 5,25, 125 etc.
3) its a multiple of all the prime factors till 100.

Now X+1
would not be an even number.
would not have zeros at the end
would not be a multiple of any of the prime numbers till 100.

Thus there are two possibilities:
1) Either X+1 is a prime number
2) X+1 is a composite number.

Now for 1) there's no option to proove its validity.
For 2) the only option one can pick up is e. Coz' from the above we can see that X+1 would not be a multiple of any number from 2 to 100.
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Joined: 17 Jun 2008
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13 Aug 2008, 23:19
Any mathematical way to get the answer? I tried the lengthy way of finding pattern...but unsuccessful.

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) greather than 40
Attachments

Doc1.doc [62.5 KiB]

Manager
Joined: 15 Jul 2008
Posts: 206
Re: GMATPresp: Smallest Prime Factor [#permalink]

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14 Aug 2008, 04:15
scthakur wrote:
Any mathematical way to get the answer? I tried the lengthy way of finding pattern...but unsuccessful.

PS take the trouble of typing the question mate..

Anyways..

h(100) = 2*4*6*....*100 = 2^50 * 50!

Now, a general rule. If x>1 is a factor of a number n, then x will not a factor of (n+1). This can be tried out with a few examples.
Factors of 15 are 3,5,15. None of these will be factors of 16. Factors of 16 are 2,4,8,16. None of these are factors of 17.. so on and so forth.

So none of the numbers 1-50 will be a factor of h(100)+1. The prime factor has to be greater than 50. So E
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Posts: 402
Re: GMATPresp: Smallest Prime Factor [#permalink]

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14 Aug 2008, 16:16
h(100)= 2*4*6*8*....*100 = 2^50(1*2*3*4*5*6* ... *50) = 2^50*50!

Notice that h(100) is divisible by every integer from 1 to 50 because of the 50!

Adding one implies that you're adding a "remainder" of 1. Therefore, the result must be divisible by a number greater than 50.

E.
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Posts: 970
Re: GMATPresp: Smallest Prime Factor [#permalink]

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14 Aug 2008, 16:50
It's amazing how you guys are able to see this:

h(100)= 2*4*6*8*....*100 = 2^50(1*2*3*4*5*6* ... *50) = 2^50*50!

I never would have been able to see this pattern.
Re: GMATPresp: Smallest Prime Factor   [#permalink] 14 Aug 2008, 16:50

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