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# What is the positive integer n? (1) For every integer m, the

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SVP
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What is the positive integer n? (1) For every integer m, the [#permalink]

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14 Dec 2008, 12:48
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What is the positive integer n?

(1) For every integer m, the product m (m + 1) (m + 2) ... (m + n) is divisible by 16
(2) n^2 - 9n + 20 = 0

Source: Gmat Club Forum Discussion.
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SVP
Joined: 17 Jun 2008
Posts: 1507
Re: DS: What is the positive integer n? [#permalink]

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14 Dec 2008, 12:53
I will go with E.

From stmt1: n = 4 or 5 (depending upon whether m is even or odd).
From stmt2: n = 4 or 5.
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Re: DS: What is the positive integer n? [#permalink]

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14 Dec 2008, 13:05
It is little more difficult than that.
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Manager
Joined: 23 Nov 2008
Posts: 79
Re: DS: What is the positive integer n? [#permalink]

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14 Dec 2008, 13:30
1
KUDOS
C for me.

From statement 1:
For every integer m, the product m (m + 1) (m + 2) ... (m + n) is divisible by 16

In a sequence of consequetive numbers, every other number will be even, and every fourth number will be a multiple of 4, so the minimum numbers you need for the products to be divisible by 16 (2^4) would be 5 (if the first is even) or 6 (if the first is odd).Since the statement is true for every integer m, (irrespective of odd or even), the minimum is 6 => minimum n = 5. But n could also be any number greater than 5 for the above to hold true, hence (1) in itself is INSUFF

Statement 2 says n is 4 or 5 - INSUFF

Together, only 5 will work, so SUFF.
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Re: DS: What is the positive integer n? [#permalink]

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15 Dec 2008, 05:41
I did not understand it all. Can someone explain the solution of twilight in layman's term?
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Re: DS: What is the positive integer n? [#permalink]

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15 Dec 2008, 06:15
statement 1 you clearly cant estimate without somewhere to start, so go to statement 2.. factoring easily you get n=4 or n=5..not suff

go back to statement 1 and plug 4 and 5 in for N. it says any integer will work, so if you just use 1 you get (5*4*3*2*1) = not divs by 16, and (6*5*4*3*2*1)=720= 16*45...so 1 and 2 together are suff.
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Re: DS: What is the positive integer n? [#permalink]

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15 Dec 2008, 23:36
scthakur, can u pls explain hw did u get n=4 or 5 in stmt1?
scthakur wrote:
I will go with E.

From stmt1: n = 4 or 5 (depending upon whether m is even or odd).
From stmt2: n = 4 or 5.
Director
Joined: 01 Apr 2008
Posts: 847
Name: Ronak Amin
Schools: IIM Lucknow (IPMX) - Class of 2014
Re: DS: What is the positive integer n? [#permalink]

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16 Dec 2008, 02:28
I Dont think we can conclude anything from statement 1.
the sequence can be:
2,3,4,5,6 => 5 terms
8,9,10 => 3 terms

From statement 2 we get 4 or 5.

Then only when we apply statement 2 to 1 we get C.
We CANNOT get 4 or 5 from statement 1. If anyone has any explaination please post.
SVP
Joined: 17 Jun 2008
Posts: 1507
Re: DS: What is the positive integer n? [#permalink]

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16 Dec 2008, 04:51
ritula wrote:
scthakur, can u pls explain hw did u get n=4 or 5 in stmt1?
scthakur wrote:
I will go with E.

From stmt1: n = 4 or 5 (depending upon whether m is even or odd).
From stmt2: n = 4 or 5.

Every third even integer is divisible by 4.
Hence, if m is an odd integer then, m+1, m+3 and m+5 will be even and one of them will also be divisible by 4. But this means, n = 5.

However, if m is even then, m, m+2 and m+4 will be even and one of them will be divisible by 4. Hence, n = 4.
Director
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Re: DS: What is the positive integer n? [#permalink]

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16 Dec 2008, 05:10
Hi scthakur,

I think you mean one in every three consecutive even integers is a multiple of 4 ( and not every third even integer is a multiple of four ).

My point is why do we take 4 terms min..8,9,10 are also consecutive and also satisfy the condition!! is it because the question says m ( m+1) (m+2)...(m+n) ??
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Re: DS: What is the positive integer n? [#permalink]

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16 Dec 2008, 05:24
Economist wrote:
Hi scthakur,

I think you mean one in every three consecutive even integers is a multiple of 4 ( and not every third even integer is a multiple of four ).

My point is why do we take 4 terms min..8,9,10 are also consecutive and also satisfy the condition!! is it because the question says m ( m+1) (m+2)...(m+n) ??

You are right with your example. I took the worst case. If we start with smallest even integer, a minimum of four consecutive even integers will be required for the product of these to be divisible by 16. Same goes for starting integer as odd integer too.

Else, 15*16 is divisible by 16 and n = 1.
16 is divisible by 16 and n = 0, etc.
Senior Manager
Joined: 21 Apr 2008
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Location: Motortown
Re: DS: What is the positive integer n? [#permalink]

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16 Dec 2008, 19:03
Why can't m be a negative number and the product, which is equal to ZERO is still divisible by 16 ?
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Re: DS: What is the positive integer n? [#permalink]

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16 Dec 2008, 20:01
I will go with C..and N=5.

see for EVERY integer m..(m=0 is just one example, if m=1 then N=4 will not work).. you need a minimum of n=5 i.e 2^4 in the sequence..
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Re: DS: What is the positive integer n? [#permalink]

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18 Dec 2008, 04:11
LiveStronger wrote:
Why can't m be a negative number and the product, which is equal to ZERO is still divisible by 16 ?

m can be a negative integer or can even be zero.

However, n = 5 is the number that will cover any value of m (whether positive, negative, zero, etc.)
SVP
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Re: DS: What is the positive integer n? [#permalink]

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18 Dec 2008, 04:11
LiveStronger wrote:
Why can't m be a negative number and the product, which is equal to ZERO is still divisible by 16 ?

m can be a negative integer or can even be zero.

However, n = 5 is the number that will cover any value of m (whether positive, negative, zero, etc.)
Senior Manager
Joined: 21 Apr 2008
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Location: Motortown
Re: DS: What is the positive integer n? [#permalink]

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18 Dec 2008, 05:15
scthakur:

I am still confused how about when m= -1 and n= 3,
m (m + 1) (m + 2) ... (m + n) is still divisible by 16
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Re: DS: What is the positive integer n? [#permalink]

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18 Dec 2008, 12:38
LiveStronger wrote:
scthakur:

I am still confused how about when m= -1 and n= 3,
m (m + 1) (m + 2) ... (m + n) is still divisible by 16

Thats a case for some value of m not for all values of m.

If m = 0, n could even be 1 but 1 doesnot work for all m. Similarly, n = 3 or 4 also donot work for all values of m. If n = 5, no matter the integer value of m, m (m + 1) (m + 2) ... (m + n) is divisible by 16.

Therefore, C
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Re: DS: What is the positive integer n?   [#permalink] 18 Dec 2008, 12:38
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