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Does P have a factor X where 1<X<P , and X and P are positive integers

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Does P have a factor X where 1<X<P , and X and P are positive integers [#permalink]

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Does P have a factor X where 1 < X < P, and X and P are positive integers?

(1) GCD (P^2, k) = k, where k is a prime number

(2) 36*20 + 2 < P < 36*20+6
[Reveal] Spoiler: OA

Last edited by Bunuel on 19 Nov 2014, 04:25, edited 1 time in total.
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Re: Does P have a factor X where 1<X<P , and X and P are positive integers [#permalink]

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Does P have a factor X where 1 < X < P, and X and P are positive integers?

The question basically asks whether p is a prime number. If it is, then it won't have a factor x such that 1 < x < p (definition of a prime number).

(1) GCD (P^2, k) = k, where k is a prime number. Can p be a prime? Yes, consider p = k = 2. Can p be a non-prime? Yes, consider p = 4 and k = 2. Not sufficient.

(2) 36*20 + 2 < P < 36*20+6 --> p could be 36*20 + 3 = 723, 36*20 + 4 = 724, or 36*20 + 5 = 725. None of these numbers is prime: 723 is a multiple of 3 (sum of its digits is a multiple of 3), 724 is even and 725 is a multiple of 5. So, we can give a definite answer that p is not a prime number, therefore it must have a factor which is greater than 1 and less than p itself. Sufficient.

Answer: B.
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Re: Does P have a factor X where 1<X<P , and X and P are positive integers [#permalink]

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Bunuel wrote:
Does P have a factor X where 1 < X < P, and X and P are positive integers?

The question basically asks whether p is a prime number. If it is, then it won't have a factor x such that 1 < x < p (definition of a prime number).

(1) GCD (P^2, k) = k, where k is a prime number. Can p be a prime? Yes, consider p = k = 2. Can p be a non-prime? Yes, consider p = 4 and k = 2. Not sufficient.

(2) 36*20 + 2 < P < 36*20+6 --> p could be 36*20 + 3 = 723, 36*20 + 4 = 724, or 36*20 + 5 = 725. None of these numbers is prime: 723 is a multiple of 3 (sum of its digits is a multiple of 3), 724 is even and 725 is a multiple of 5. So, we can give a definite answer that p is not a prime number, therefore it must have a factor which is greater than 1 and less than p itself. Sufficient.

Answer: B.


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Re: Does P have a factor X where 1<X<P , and X and P are positive integers [#permalink]

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New post 15 Nov 2015, 09:37
Bunuel wrote:
Does P have a factor X where 1 < X < P, and X and P are positive integers?

The question basically asks whether p is a prime number. If it is, then it won't have a factor x such that 1 < x < p (definition of a prime number).

(1) GCD (P^2, k) = k, where k is a prime number. Can p be a prime? Yes, consider p = k = 2. Can p be a non-prime? Yes, consider p = 4 and k = 2. Not sufficient.

(2) 36*20 + 2 < P < 36*20+6 --> p could be 36*20 + 3 = 723, 36*20 + 4 = 724, or 36*20 + 5 = 725. None of these numbers is prime: 723 is a multiple of 3 (sum of its digits is a multiple of 3), 724 is even and 725 is a multiple of 5. So, we can give a definite answer that p is not a prime number, therefore it must have a factor which is greater than 1 and less than p itself. Sufficient.

Answer: B.


but what about X itself. for example, if X=721 and in this case it is not a factor of P?

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Re: Does P have a factor X where 1<X<P , and X and P are positive integers [#permalink]

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New post 15 Nov 2015, 22:59
vadam wrote:
Bunuel wrote:
Does P have a factor X where 1 < X < P, and X and P are positive integers?

The question basically asks whether p is a prime number. If it is, then it won't have a factor x such that 1 < x < p (definition of a prime number).

(1) GCD (P^2, k) = k, where k is a prime number. Can p be a prime? Yes, consider p = k = 2. Can p be a non-prime? Yes, consider p = 4 and k = 2. Not sufficient.

(2) 36*20 + 2 < P < 36*20+6 --> p could be 36*20 + 3 = 723, 36*20 + 4 = 724, or 36*20 + 5 = 725. None of these numbers is prime: 723 is a multiple of 3 (sum of its digits is a multiple of 3), 724 is even and 725 is a multiple of 5. So, we can give a definite answer that p is not a prime number, therefore it must have a factor which is greater than 1 and less than p itself. Sufficient.

Answer: B.


but what about X itself. for example, if X=721 and in this case it is not a factor of P?


Your question is not clear. First of all the question asks "Does P have a factor X where 1 < X < P..." plus, an integer is a factor of itself. For example, 721 is a factor of 721.
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Re: Does P have a factor X where 1<X<P , and X and P are positive integers [#permalink]

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New post 09 Feb 2016, 18:34
manpreetsingh86 wrote:
Does P have a factor X where 1 < X < P, and X and P are positive integers?


(1) GCD (P^2, k) = k, where k is a prime number
from this, we know that k is a factor of p^2. but we don't know whether k is a factor of P or not. what if k=p? what if k=p=1? p^2=1. since we need integers, there might be no x at all here.
suppose we have p=9
k=3. then yes. it might be a number x that is a factor of p.


(2) 36*20 + 2 < P < 36*20+6
722 < P < 726
p can be 723, 724, 725.
724 is even, so it is divisible
725 ends in 5 so is divisible by 5
7+2+3=12, so divisible by 3
we see that all possible options of P have other factors. we can give a definite answer.
B.

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Re: Does P have a factor X where 1<X<P , and X and P are positive integers [#permalink]

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New post 16 Mar 2016, 04:05
Here all the values between 36*20+2 and 36*20+6 are non primes as it it possible to obtain the factors.
Hence B is sufficient
statement 2 can be discarded as p can be 3 and p can be 81 (for k=3)
thus B is sufficient
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Re: Does P have a factor X where 1<X<P , and X and P are positive integers [#permalink]

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New post 08 Oct 2017, 16:47
Bunuel wrote:
Does P have a factor X where 1 < X < P, and X and P are positive integers?

The question basically asks whether p is a prime number. If it is, then it won't have a factor x such that 1 < x < p (definition of a prime number).

(1) GCD (P^2, k) = k, where k is a prime number. Can p be a prime? Yes, consider p = k = 2. Can p be a non-prime? Yes, consider p = 4 and k = 2. Not sufficient.

(2) 36*20 + 2 < P < 36*20+6 --> p could be 36*20 + 3 = 723, 36*20 + 4 = 724, or 36*20 + 5 = 725. None of these numbers is prime: 723 is a multiple of 3 (sum of its digits is a multiple of 3), 724 is even and 725 is a multiple of 5. So, we can give a definite answer that p is not a prime number, therefore it must have a factor which is greater than 1 and less than p itself. Sufficient.

Answer: B.


Hi Bunuel..

How did u conclude that P is a prime number?

Pls help..thanks in advance

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Re: Does P have a factor X where 1<X<P , and X and P are positive integers [#permalink]

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New post 08 Oct 2017, 20:01
zanaik89 wrote:
Bunuel wrote:
Does P have a factor X where 1 < X < P, and X and P are positive integers?

The question basically asks whether p is a prime number. If it is, then it won't have a factor x such that 1 < x < p (definition of a prime number).

(1) GCD (P^2, k) = k, where k is a prime number. Can p be a prime? Yes, consider p = k = 2. Can p be a non-prime? Yes, consider p = 4 and k = 2. Not sufficient.

(2) 36*20 + 2 < P < 36*20+6 --> p could be 36*20 + 3 = 723, 36*20 + 4 = 724, or 36*20 + 5 = 725. None of these numbers is prime: 723 is a multiple of 3 (sum of its digits is a multiple of 3), 724 is even and 725 is a multiple of 5. So, we can give a definite answer that p is not a prime number, therefore it must have a factor which is greater than 1 and less than p itself. Sufficient.

Answer: B.


Hi Bunuel..

How did u conclude that P is a prime number? The question asks whether p is prime. If p is prime it won't have a factor x such that 1 < x < p, if p is NOT prime it will have a factor x such that 1 < x < p. For example, prime number 7, does not have a factor x such that 1 < x < 7.

Pls help..thanks in advance


Where does it say that p is a prime?
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Re: Does P have a factor X where 1<X<P , and X and P are positive integers   [#permalink] 08 Oct 2017, 20:01
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