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# If p is a positive integer and 10p/96 is an integer

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Intern
Joined: 17 Dec 2012
Posts: 28
If p is a positive integer and 10p/96 is an integer  [#permalink]

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29 Jul 2014, 13:45
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Question Stats:

69% (01:18) correct 31% (01:33) wrong based on 250 sessions

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If p is a positive integer and 10p/96 is an integer, then the minimum number of prime factors p could have is

(A) One
(B) Two
(C) Three
(D) Four
(E) Five
Math Expert
Joined: 02 Sep 2009
Posts: 57155
Re: If p is a positive integer and 10p/96 is an integer  [#permalink]

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29 Jul 2014, 13:51
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1
irda wrote:
If p is a positive integer and 10p/96 is an integer, then the minimum number of prime factors p could have is

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

10p/96 = 5p/48.

The least positive value of p for which 5p/48 is an integer is 48 = 2^3*3. Hence the minimum number of prime factors p could have is two, namely 2 and 3.

Hope it's clear.
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Joined: 17 Dec 2012
Posts: 28
Re: If p is a positive integer and 10p/96 is an integer  [#permalink]

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29 Jul 2014, 14:06
1
Bunuel,

Thanks. The logic behind the question is unacceptable per my chain of thoughts. I am definitely wrong as both your answer and the answer from the source match, but how can we state the value of unique prime numbers as the value of minimum prime numbers.

Does not minimum number of prime indicate the total number of prime numbers that must be in the numerator? The question does not say unique prime.

can you please help dispel my confusion. It is painful to override my flawed logic , if so.

and thanks again
Math Expert
Joined: 02 Sep 2009
Posts: 57155
Re: If p is a positive integer and 10p/96 is an integer  [#permalink]

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29 Jul 2014, 14:23
irda wrote:
Bunuel,

Thanks. The logic behind the question is unacceptable per my chain of thoughts. I am definitely wrong as both your answer and the answer from the source match, but how can we state the value of unique prime numbers as the value of minimum prime numbers.

Does not minimum number of prime indicate the total number of prime numbers that must be in the numerator? The question does not say unique prime.

can you please help dispel my confusion. It is painful to override my flawed logic , if so.

and thanks again

I see your point but usually the number of prime factors means the number of unique prime factors. For example, we say that 8 has one prime factor - 2, not three primes 2, 2, and 2. Though on the real test, I think, this would be explicitly stated.
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Intern
Joined: 17 Dec 2012
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Re: If p is a positive integer and 10p/96 is an integer  [#permalink]

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29 Jul 2014, 14:31
1
Your assertion has such remedial effect to us novices. I must accept it is much more than just placebo. Thanks dude.
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Joined: 22 Feb 2009
Posts: 158
Re: If p is a positive integer and 10p/96 is an integer  [#permalink]

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29 Jul 2014, 18:49
Bunuel wrote:
irda wrote:
If p is a positive integer and 10p/96 is an integer, then the minimum number of prime factors p could have is

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

10p/96 = 5p/48.

The least positive value of p for which 5p/48 is an integer is 48 = 2^3*3. Hence the minimum number of prime factors p could have is two, namely 2 and 3.

Hope it's clear.

yeah, it is totally clear
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Re: If p is a positive integer and 10p/96 is an integer  [#permalink]

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03 Apr 2017, 06:01
96= 2^5*3^1

(2*5*p)/2^5*3

Therefore p= 2^4*3^1 (At least)

Two prime factors.
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Intern
Joined: 18 Jul 2018
Posts: 36
Re: If p is a positive integer and 10p/96 is an integer  [#permalink]

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23 Jan 2019, 02:23
Bunuel wrote:
irda wrote:
Bunuel,

Thanks. The logic behind the question is unacceptable per my chain of thoughts. I am definitely wrong as both your answer and the answer from the source match, but how can we state the value of unique prime numbers as the value of minimum prime numbers.

Does not minimum number of prime indicate the total number of prime numbers that must be in the numerator? The question does not say unique prime.

can you please help dispel my confusion. It is painful to override my flawed logic , if so.

and thanks again

I see your point but usually the number of prime factors means the number of unique prime factors. For example, we say that 8 has one prime factor - 2, not three primes 2, 2, and 2. Though on the real test, I think, this would be explicitly stated.

So should we always consider "number of prime factors" as "nunber of unique primes"...? I am a bit confused as those2 terms are totally 2 different concepts.
Math Expert
Joined: 02 Sep 2009
Posts: 57155
Re: If p is a positive integer and 10p/96 is an integer  [#permalink]

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23 Jan 2019, 02:26
iac00 wrote:
Bunuel wrote:
irda wrote:
Bunuel,

Thanks. The logic behind the question is unacceptable per my chain of thoughts. I am definitely wrong as both your answer and the answer from the source match, but how can we state the value of unique prime numbers as the value of minimum prime numbers.

Does not minimum number of prime indicate the total number of prime numbers that must be in the numerator? The question does not say unique prime.

can you please help dispel my confusion. It is painful to override my flawed logic , if so.

and thanks again

I see your point but usually the number of prime factors means the number of unique prime factors. For example, we say that 8 has one prime factor - 2, not three primes 2, 2, and 2. Though on the real test, I think, this would be explicitly stated.

So should we always consider "number of prime factors" as "nunber of unique primes"...? I am a bit confused as those2 terms are totally 2 different concepts.

Yes. I think I answered this above.
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Re: If p is a positive integer and 10p/96 is an integer  [#permalink]

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23 Jan 2019, 07:25
irda wrote:
If p is a positive integer and 10p/96 is an integer, then the minimum number of prime factors p could have is

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

10p/96
96= 2^ 5*3^1

or say
5p/48
or 5p/2^4 *3

so p has to have atleast 2 factors of 2 & 3 ; IMO B
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Re: If p is a positive integer and 10p/96 is an integer  [#permalink]

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27 Jan 2019, 19:36
irda wrote:
If p is a positive integer and 10p/96 is an integer, then the minimum number of prime factors p could have is

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

Simplifying we have:

10p/96 = 5p/48

In order for 5p/48 to be an integer, then p must be a multiple of 48. Since 48 = 2^4 x 3, we see that the minimum number of prime factors of p is 2.

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Re: If p is a positive integer and 10p/96 is an integer   [#permalink] 27 Jan 2019, 19:36
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