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# If P represents the product of the first 15 positive integers, then P

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If P represents the product of the first 15 positive integers, then P  [#permalink]

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13 Mar 2018, 01:30
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5% (low)

Question Stats:

94% (01:11) correct 6% (00:34) wrong based on 77 sessions

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If P represents the product of the first 15 positive integers, then P is not a multiple of:

A. 99
B. 84
C. 72
D. 65
E. 57

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Re: If P represents the product of the first 15 positive integers, then P  [#permalink]

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13 Mar 2018, 04:05
Bunuel wrote:
If P represents the product of the first 15 positive integers, then P is not a multiple of:

A. 99
B. 84
C. 72
D. 65
E. 57

P = 15! as per definition of question

A. 99 = 9*11 Yes 15! has both factors
B. 84 = 3*4*7 Yes 15! has both factors
C. 72 = 9*8 Yes 15! has both factors
D. 65 = 9*13 Yes 15! has both factors
E. 57 = 3*19 NO 15! DOES NOT HAVE A FACTOR 19 hence CORRECT ANSWER

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Re: If P represents the product of the first 15 positive integers, then P  [#permalink]

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14 Mar 2018, 16:10
Bunuel wrote:
If P represents the product of the first 15 positive integers, then P is not a multiple of:

A. 99
B. 84
C. 72
D. 65
E. 57

Let’s look at the answer choices.

Since 99 = 9 x 11, both of which are included in the product of the first 15 integers, we see that P is a multiple of 99.

Similarly, since 84 = 42 x 2 = 21 x 4 = 7 x 3 x 4, we see that P is a multiple of 84.

Since 72 = 36 x 2 = 6 x 3 x 2 x 2 = 6 x 3 x 4, we see that P is a multiple of 72.

Since 65 = 5 x 13, we see that P is a multiple of 65.

Since 57 = 19 x 3, and since the product of the first 15 integers does not have a prime factor of 19, P cannot be a multiple of 57.

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Re: If P represents the product of the first 15 positive integers, then P  [#permalink]

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19 Mar 2018, 10:57

SOLUTION

P= 1*2*3*4*5*6*7*8*9*10*11*12*13*14*15
For P to be a multiple of some number, P should be completely divisible by the same number.

Now let us find the option which does not divide P.

• 99= 9*11
o Since P contains 9*11, hence P is divisible by 99.

• 84=7*12
o Since P contains 7*12, hence P is divisible by 84.

• 72=8*9
o Since P contains 8*9, hence P is divisible by 72.

• 65=5*13
o Since P contains 5*13, hence P is divisible by 65.

• 57=3*19
o Since P does not contain 19, hence P is not divisible by 57.

Hence, 57 is not a multiple of P.

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Re: If P represents the product of the first 15 positive integers, then P  [#permalink]

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19 Mar 2018, 13:30
Lets have a look at the answer choices-
A. 99 - 9X11- both are in P
B. 84 - 7X12- both are in P
C. 72 - 7X8- both are in P
D. 65 - 5X13- both are in P
E. 57 - 3X19

19 is a prime number and not there in the product- P.
Re: If P represents the product of the first 15 positive integers, then P   [#permalink] 19 Mar 2018, 13:30
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