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# If a positive odd integer N has p positive factors, how many positive

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Current Student
Joined: 12 Aug 2015
Posts: 2621
Schools: Boston U '20 (M)
GRE 1: Q169 V154
If a positive odd integer N has p positive factors, how many positive  [#permalink]

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Updated on: 09 Aug 2016, 07:21
3
00:00

Difficulty:

55% (hard)

Question Stats:

58% (01:24) correct 42% (01:26) wrong based on 153 sessions

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If a positive odd integer N has p positive factors, how many positive factors will 2N have ?

A) p
B) 2p
C) p + 1
D) 2p + 1
E) Cannot be determined

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Originally posted by stonecold on 07 Aug 2016, 20:08.
Last edited by stonecold on 09 Aug 2016, 07:21, edited 1 time in total.
Current Student
Joined: 22 Jun 2016
Posts: 242
Re: If a positive odd integer N has p positive factors, how many positive  [#permalink]

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07 Aug 2016, 22:41
3
no of factor of a=b^m * c^n * d^o * e^q.... = (m+1)*(n+1)*(o+1)*(q+1)...

Given that N is an odd integer. So N can be b^m * c^n * d^o * e^q, where b,c,d,e are odd numbers.
The number of factors of N = (m+1)*(n+1)*(o+1)*(q+1) = p

For a number 2N = 2^1 * b^m * c^n * d^o * e^p, number of factors will be (1+1)*(m+1)*(n+1)*(o+1)*(q+1) = 2*p

So, number of factors of 2N = 2*p

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Re: If a positive odd integer N has p positive factors, how many positive  [#permalink]

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09 Aug 2016, 07:16
2
Top Contributor
I have added some extra text to make this question more GMAT-like:

stonecold wrote:
if positive odd integer N has p positive factors, how many positive factors will 2N have ?
A) p
B) 2p
C) P+1
D) 2p+1
E) Cannot be determined

Let's TEST some values of N
Try N = 3
The factors of 3 are {1, 3}. Here, p = 2
So, 2N = (2)(3) = 6
The factors of 6 are {1, 2, 3, 6}. So, we have a total of 4

A) p = 2 No good. We want an output of 4. ELIMINATE
B) 2p = (2)(2) = 4. PERFECT! KEEP B
C) P+1 = 2 + 1 = 3 No good. We want an output of 4. ELIMINATE
D) 2p+1 = (2)(2) + 1 = 5 No good. We want an output of 4. ELIMINATE
E) Cannot be determined. POSSIBLE. KEEP E

Let's TEST another value of N
Try N = 7
The factors of 7 are {1, 7}. Here, p = 2
So, 2N = (2)(7) = 14
The factors of 14 are {1, 2, 7, 14}. So, we have a total of 4

Now check the REMAINING answer choices:
B) 2p = (2)(2) = 4. PERFECT! KEEP B
E) Cannot be determined. POSSIBLE. KEEP E

Let's TEST one more (non-prime) value of N
Try N = 9
The factors of 9 are {1, 3, 9}. Here, p = 3
So, 2N = (2)(9) = 18
The factors of 18 are {1, 2, 3, 6, 9}. So, we have a total of 6

Now check the REMAINING answer choices:
B) 2p = (2)(3) = 6. PERFECT! KEEP B
E) Cannot be determined. POSSIBLE. KEEP E

At this point, it SEEMS LIKELY that the correct answer is B
NOTE: This strategy of testing values, while not perfect, can still help you eliminate answer choices to give yourself a better chance of correctly guessing the right answer.

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Current Student
Joined: 12 Aug 2015
Posts: 2621
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Re: If a positive odd integer N has p positive factors, how many positive  [#permalink]

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15 Dec 2016, 04:49
Here dismay understanding of this Question=>

Firstly,Any odd integer can never have any even divisor.
Now N is an odd integer with => P factors.
Each of these P factors is odd.

For 2N => Each Odd factor will have its own even Divisor counterpart.
E.g => 9 has 3 divisors =>
1
3
9

2N=18 => Divisors will be =>
1
1*2=2
3
3*2=6
9
9*2=18

Hence 2N will have 2p factors in total.

Hence B
A few Takeaways from this Question=>
If N is odd and has P factors => 2N will have 2P Factors.
One must be a factor of every number ,so if the Question says that the difference between any of its factors is even => All factors must be odd.So, the number must be odd
.

NOTE => If n is even we cannot say that 2N will have 2p factors.
E.g =>
2 => 2 factors
4 => 3 factors

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Re: If a positive odd integer N has p positive factors, how many positive  [#permalink]

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22 Dec 2017, 06:26
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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If a positive odd integer N has p positive factors, how many positive   [#permalink] 22 Dec 2017, 06:26
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