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# Can the positive integer p be expressed as the product of

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Can the positive integer p be expressed as the product of  [#permalink]

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27 Dec 2009, 05:49
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Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?

1) 31 < p < 37
2) p is odd
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27 Dec 2009, 07:24
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Minotaur wrote:
Can the positive integer p be expressed as the product of two integers, each of which is greater than 1 ?

(1) 31 < p < 37
(2) p is odd

OA:

can anyone help me with the explanation to this problem.

The wording makes this question harder than it is actually.

If positive integer p cannot be expressed as the product of two integers >1, it simply means that p is a prime number. So, basically question asks is p prime?

(1) 31<p<37 --> between these numbers there is no prime. Hence ANY integer from these range CAN be expresses as the product of two numbers. Sufficient.

(2) p is odd --> odd numbers can be primes as well as non-primes. Not sufficient.

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27 Dec 2009, 08:27
s1)----> p can be 32,33,34,35,36 each no. is having at least 2 factors >1 hence yes .....therfore suff

s2)----> p can be any odd int and for odd like 3 , 5,7 ans is no and for no's such as 15,21,27, ans is yes
hence insuff
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17 Jun 2010, 07:45
Thanks for the explanation Buunel and xcusemeplz2009.
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08 Jul 2011, 20:51
1. Sufficient

p can be 32 or 33 or 34 or 35 or 36

and each of these numbers can be expressed as product of two integers that are >1 . (32 = 2*16 , 33 = 3*11....)

2. Not sufficient

p is odd

p = 1 3 5 15

when p= 1 or 3 or 5 it cannot be expressed as product of two integers greater than 1. (as 5 = 1*5)

when p=15 , p can be expressed as product of two integers greater than 1.

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25 Aug 2011, 13:38
1
Bunuel,

This question confuses me big time. When I saw "CAN" I expected as sufficient to be able to prove that yes, there was a way to do so. Hence for statement (2) I'd say that yes, IT CAN BE EXPRESSED AS THE PRODUCT OF TWO INTEGERS. Do you consider the following two statements to have the same meaning?

(i) Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?

(ii) Is the positive integer p the product of two integers, each of which is greater than 1?
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16 Sep 2011, 10:16
Bunuel wrote:
Minotaur wrote:
Can the positive integer p be expressed as the product of two integers, each of which is greater than 1 ?

(1) 31 < p < 37
(2) p is odd

OA:

can anyone help me with the explanation to this problem.

The wording makes this question harder than it is actually.

If positive integer p can not be expressed as the product of two integers >1, it simply means that p is a prime number. So, basically question asks is p prime?

(1) 31<p<37 --> between these numbers there is no prime. Hence ANY integer from these range CAN be expresses as the product of two numbers. Sufficient.

(2) p is odd --> odd numbers can be primes as well as non-primes. Not sufficient.

If positive integer p can not be expressed as the product of two integers >1, it simply means that p is a prime number. So, basically question asks is p prime?
how can u say this..please elaborate...how p is prime...any no greater than one can be any no..why r u sayin p is prime?
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16 Sep 2011, 11:26
ruturaj wrote:
If positive integer p can not be expressed as the product of two integers >1, it simply means that p is a prime number. So, basically question asks is p prime?
how can u say this..please elaborate...how p is prime...any no greater than one can be any no..why r u sayin p is prime?

Prime number can only be expressed as "1*p", where p is the prime number itself

13=1*13

Can we write any prime number in the form;
p=m*n
where,
p=prime number
m=integer greater than 1
n=integer greater than 1
No, right? For prime number, at least one of m and n must be 1.

Thus, question is indirectly asking whether p is a prime number.
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Re: Can the positive integer p be expressed as the product of  [#permalink]

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23 Feb 2012, 21:08
1
I don't understand this question. I am getting E.

Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?

Statement (1) states that 31<p<37

My logic was that p can be anything that fits between 31 and 37, so 36
36 = 9 x 4 and each integer is greater than 1, so it is sufficient
36 = -9 x -4 and each integer is not greater than 1, so insufficient.

Statement (2), I agree it is insufficient.

(1) and (2) is still insufficient because 35 (fits statement 2) and 35 can be 7x5 so each integer is greater than 1. 35 can also be -7x-5 so each integer is not greater than 1.

Am I misunderstanding the question?
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Posts: 47967
Re: Can the positive integer p be expressed as the product of  [#permalink]

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24 Feb 2012, 01:25
1
chamisool wrote:
I don't understand this question. I am getting E.

Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?

Statement (1) states that 31<p<37

My logic was that p can be anything that fits between 31 and 37, so 36
36 = 9 x 4 and each integer is greater than 1, so it is sufficient
36 = -9 x -4 and each integer is not greater than 1, so insufficient.

Statement (2), I agree it is insufficient.

(1) and (2) is still insufficient because 35 (fits statement 2) and 35 can be 7x5 so each integer is greater than 1. 35 can also be -7x-5 so each integer is not greater than 1.

Am I misunderstanding the question?

It seems that you misinterpreted the question.

Look at the definition of a prime number: a prime number is a positive integer with exactly two factors: 1 and itself. Now, the questions asks: "can the positive integer p be expressed as the product of two integers, each of which is greater than 1" So, the question basically asks whether p is a prime number, because if it is then p can NOT be expressed as the product of two integers, each of which is greater than 1.

(1) states: 31 < p < 37. Between these numbers there is no prime. Hence ANY integer from these range CAN be expresses as the product of two numbers, which means that the answer to the question is YES: p can always be expressed as the product of two integers, each of which is greater than 1. Sufficient.

Just to illustrate: 32=2*18, 33=3*11, 34=2*17, 35=5*7, 36=2*18.

Hope it's clear.
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Can the positive integer P be expressed as a product of 2...  [#permalink]

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28 Feb 2012, 15:28
Can the positive integer P be expressed as a product of two integers,each of which is greater than 1?
(1) 31<p<37
(2) p=odd

The answer according my program is A, but I dont understand why it can not be D. Because if we take 3*3=9 which is odd and integer and greater than 1?

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Posts: 47967
Re: Can the positive integer P be expressed as a product of 2...  [#permalink]

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28 Feb 2012, 15:35
Merging similar topics.

Can the positive integer P be expressed as a product of two integers, each of which is greater than 1?
(1) 31<p<37
(2) p=odd

The answer according my program is A, but I dont understand why it can not be D. Because if we take 3*3=9 which is odd and integer and greater than 1?

P is some particular integer and we are asked whether it can be expressed as a product of two integers, each of which is greater than 1. Now, for (2) if p=9 then the answer is YES, it can be expressed as a product of two integers, each of which is greater than 1 but of p=5 then the answer is NO, it cannot be expressed as a product of two integers, each of which is greater than 1. Two different answers, hence this statement is not sufficient.

Does it makes sense?

P.S. Please refer for a complete solution to the above posta and ask if anything remains unclear.
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Re: Can the positive integer P be expressed as a product of 2...  [#permalink]

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28 Feb 2012, 16:51
Bunuel wrote:
Merging similar topics.

Can the positive integer P be expressed as a product of two integers, each of which is greater than 1?
(1) 31<p<37
(2) p=odd

The answer according my program is A, but I dont understand why it can not be D. Because if we take 3*3=9 which is odd and integer and greater than 1?

P is some particular integer and we are asked whether it can be expressed as a product of two integers, each of which is greater than 1. Now, for (2) if p=9 then the answer is YES, it can be expressed as a product of two integers, each of which is greater than 1 but of p=5 then the answer is NO, it cannot be expressed as a product of two integers, each of which is greater than 1. Two different answers, hence this statement is not sufficient.

Does it makes sense?

P.S. Please refer for a complete solution to the above posta and ask if anything remains unclear.

Thank you very much Bunuel,

So basically if there are 2 possible answers (yes/no) it will always be insufficient?
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Re: Can the positive integer P be expressed as a product of 2...  [#permalink]

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28 Feb 2012, 16:55
1
Thank you very much Bunuel,

So basically if there are 2 possible answers (yes/no) it will always be insufficient?

It's a YES/NO DS question. In a Yes/No Data Sufficiency question, statement is sufficient if the answer is “always yes” or “always no” while a statement is insufficient if the answer is "sometimes yes" and "sometimes no".
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Re: Can the positive integer p be expressed as the product of  [#permalink]

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08 Apr 2012, 05:51
Bunuel wrote:
chamisool wrote:
I don't understand this question. I am getting E.

Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?

Statement (1) states that 31<p<37

My logic was that p can be anything that fits between 31 and 37, so 36
36 = 9 x 4 and each integer is greater than 1, so it is sufficient
36 = -9 x -4 and each integer is not greater than 1, so insufficient.

Statement (2), I agree it is insufficient.

(1) and (2) is still insufficient because 35 (fits statement 2) and 35 can be 7x5 so each integer is greater than 1. 35 can also be -7x-5 so each integer is not greater than 1.

Am I misunderstanding the question?

It seems that you misinterpreted the question.

Look at the definition of a prime number: a prime number is a positive integer with exactly two factors: 1 and itself. Now, the questions asks: "can the positive integer p be expressed as the product of two integers, each of which is greater than 1" So, the question basically asks whether p is a prime number, because if it is then p can NOT be expressed as the product of two integers, each of which is greater than 1.

(1) states: 31 < p < 37. Between these numbers there is no prime. Hence ANY integer from these range CAN be expresses as the product of two numbers, which means that the answer to the question is YES: p can always be expressed as the product of two integers, each of which is greater than 1. Sufficient.

Just to illustrate: 32=2*18, 33=3*11, 34=2*17, 35=5*7, 36=2*18.

Hope it's clear.

I think as you said, the key to this problem are the words each of which is greater than one
is correct ???

You think the problem can be solved even if you do not see this nuance ??

thanks
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Re: Can the positive integer p be expressed as the product of  [#permalink]

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13 Apr 2013, 08:13
1
Ah...I struggled with this one at first as well

I originally got D because I thought the question was asking if we can have product of two numbers for p.

Key for me was reminding myself that this is as "Yes or No" question, which means that it's "always yes" or "always no." For some reason, I had interpreted "Can the positive..." as is there a single instance where it can be true.

1) 31 < p < 37...
32 = 8 x 4
33 = 11 x 3
34 = 2 x 17
35 = 5 x7
36= 6 x 6
sufficient

2) p is odd
p = 5
p = 15
not sufficient

So, the answer is A because statement 2 is not ALWAYS sufficient
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Re: Can the positive integer p be expressed as the product of  [#permalink]

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12 Sep 2013, 09:53
1
chamisool wrote:
I don't understand this question. I am getting E.

Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?

Statement (1) states that 31<p<37

My logic was that p can be anything that fits between 31 and 37, so 36
36 = 9 x 4 and each integer is greater than 1, so it is sufficient
36 = -9 x -4 and each integer is not greater than 1, so insufficient.

Statement (2), I agree it is insufficient.

(1) and (2) is still insufficient because 35 (fits statement 2) and 35 can be 7x5 so each integer is greater than 1. 35 can also be -7x-5 so each integer is not greater than 1.

Am I misunderstanding the question?

Hi Bunuel,
The question asks: can p be expressed as a product of two numbers greater than 1? If I prove that p can be expressed as a product of two numbers which are < 1 and can also be expressed as a product of two numbers > 1 then the statement A) / B) would become insufficient. Right?

Now, as quoted above, for statement A), I can express 36= 1*36, -1*-36,-9*-4..... so doesn't it mean that statement A) is insufficient?

If A) is sufficient to answer the question, is it because of the fact that the question asks "Can it be expressed as product of two numbers > 1" instead of "Is P a product of two numbers which are always greater than 1?"
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Re: Can the positive integer p be expressed as the product of  [#permalink]

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12 Sep 2013, 11:18
SurabhiStar wrote:
chamisool wrote:
I don't understand this question. I am getting E.

Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?

Statement (1) states that 31<p<37

My logic was that p can be anything that fits between 31 and 37, so 36
36 = 9 x 4 and each integer is greater than 1, so it is sufficient
36 = -9 x -4 and each integer is not greater than 1, so insufficient.

Statement (2), I agree it is insufficient.

(1) and (2) is still insufficient because 35 (fits statement 2) and 35 can be 7x5 so each integer is greater than 1. 35 can also be -7x-5 so each integer is not greater than 1.

Am I misunderstanding the question?

Hi Bunuel,
The question asks: can p be expressed as a product of two numbers greater than 1? If I prove that p can be expressed as a product of two numbers which are < 1 and can also be expressed as a product of two numbers > 1 then the statement A) / B) would become insufficient. Right?

Now, as quoted above, for statement A), I can express 36= 1*36, -1*-36,-9*-4..... so doesn't it mean that statement A) is insufficient?

If A) is sufficient to answer the question, is it because of the fact that the question asks "Can it be expressed as product of two numbers > 1" instead of "Is P a product of two numbers which are always greater than 1?"

No, the red part is not correct.

The question asks "can p be expressed as the product of two integers, each of which is greater than 1".

If from a statement you get that EACH possible value of p can be expressed as the product of two integers, each of which is greater than 1, then the answer is YES, and the statement is sufficient.

If from a statement you get that NONE of the possible values of p can be expressed as the product of two integers, each of which is greater than 1, then the answer is NO, and the statement is sufficient too.

If from a statement you get that some possible values of p cannot but other possible values of p can be expressed as the product of two integers, each of which is greater than 1, then we'd have two answers to the question and the statement wouldn't be sufficient.

Hope it's clear.
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Re: Can the positive integer p be expressed as the product of  [#permalink]

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23 Sep 2013, 13:08
What if i say that P = 33 x 1.

So it implies that surely one of the integer is greater than 1 but other one is 1 itself ?

Mental block..!!!
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Re: Can the positive integer p be expressed as the product of  [#permalink]

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24 Sep 2013, 00:20
abhisheksharma wrote:
What if i say that P = 33 x 1.

So it implies that surely one of the integer is greater than 1 but other one is 1 itself ?

Mental block..!!!

33 also can be expressed as 3*11, so the answer to the question "can the positive integer P be expressed as a product of two integers, each of which is greater than 1?" is still YES.
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Re: Can the positive integer p be expressed as the product of &nbs [#permalink] 24 Sep 2013, 00:20

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