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Integers Problem

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Source: — Data Sufficiency |

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by Rahul@gurome » Fri Dec 03, 2010 2:06 am
Given: If an integer k have a factor p such that 1 < p < k, then k is a composite number, otherwise k is prime. Thus the question simply asks whether k is prime or composite.

Statement 1: k > 4!
k may be prime (29, 31 etc) or may be composite (25, 26 etc.)

Not sufficient.

Statement 2: (13! + 2) ≤ k ≤ (13! + 13)
Each possible value of k is composite integer. Take few for example,
  • (1) k = (13! + 2) = (13*12*11*10*9*8*7*6*5*4*3*2*1 + 2) = 2*(13*12*11*10*9*8*7*6*5*4*3*1 + 1) => Multiple of 2

    (2) k = (13! + 3) = (13*12*11*10*9*8*7*6*5*4*3*2*1 + 3) = 3*(13*12*11*10*9*8*7*6*5*4*2*1 + 1) => Multiple of 3

    (3) k = (13! + 4) = (13*12*11*10*9*8*7*6*5*4*3*2*1 + 4) = 4*(13*12*11*10*9*8*7*6*5*3*2*1 + 1) => Multiple of 4

    (4) Same for 5, 6, 7, 8, 9, 10, 11, 12 and 13
Sufficient.

The correct answer is B.
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by anirudhbhalotia » Fri Dec 03, 2010 5:28 am
Rahul@gurome wrote:Given: If an integer k have a factor p such that 1 < p < k, then k is a composite number, otherwise k is prime. Thus the question simply asks whether k is prime or composite.

Statement 1: k > 4!
k may be prime (29, 31 etc) or may be composite (25, 26 etc.)

Not sufficient.

Statement 2: (13! + 2) ≤ k ≤ (13! + 13)
Each possible value of k is composite integer. Take few for example,
  • (1) k = (13! + 2) = (13*12*11*10*9*8*7*6*5*4*3*2*1 + 2) = 2*(13*12*11*10*9*8*7*6*5*4*3*1 + 1) => Multiple of 2

    (2) k = (13! + 3) = (13*12*11*10*9*8*7*6*5*4*3*2*1 + 3) = 3*(13*12*11*10*9*8*7*6*5*4*2*1 + 1) => Multiple of 3

    (3) k = (13! + 4) = (13*12*11*10*9*8*7*6*5*4*3*2*1 + 4) = 4*(13*12*11*10*9*8*7*6*5*3*2*1 + 1) => Multiple of 4

    (4) Same for 5, 6, 7, 8, 9, 10, 11, 12 and 13
Sufficient.

The correct answer is B.


Not able to understand Rahul! :-(

How did you get that its about prime or composite?

" integer k have a factor p"...does this means, for example k = 8, p = 2 or 4 or 8?

Can you break it down further...or may be I am missing some fundamental concepts here?
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by Rahul@gurome » Fri Dec 03, 2010 7:15 am
anirudhbhalotia wrote:Not able to understand Rahul! :-(

How did you get that its about prime or composite?

" integer k have a factor p"...does this means, for example k = 8, p = 2 or 4 or 8?

Can you break it down further...or may be I am missing some fundamental concepts here?
"integer k have a factor p" means p can any of the factors of integer k. For example if k = 8, then the factors of k are 1, 2, 4 and 8. Thus the value of p can be either 1 or 2 or 4 or 8.

Now the question says integer k have a factor p such that 1 < p < k. This means p ≠ 1 and p ≠ k. Now if k = 8, the only possible values of p are 2 and 8.

The definition of prime number is: A prime number is a natural number that has exactly two distinct natural number divisors: 1 and itself. Which simply means if k is a natural number and p is its factor, then for k to be prime, only possible values of p are 1 and k. Otherwise it is not prime.

Thus the question boils down to whether k is prime or composite (not prime is composite).

Hope it clears your doubt.
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