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# If p is a positive integer, is 2^p + 1 a prime number?

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Math Expert
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If p is a positive integer, is 2^p + 1 a prime number?  [#permalink]

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04 Aug 2018, 09:39
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Difficulty:

35% (medium)

Question Stats:

80% (01:04) correct 20% (00:55) wrong based on 71 sessions

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If p is a positive integer, is 2^p + 1 a prime number?

(1) p is a prime number.
(2) p is an even number.

NEW question from GMAT® Quantitative Review 2019

(DS19208)

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Re: If p is a positive integer, is 2^p + 1 a prime number?  [#permalink]

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04 Aug 2018, 10:10
1
(1) p is a prime number.
2^2 + 1 = 5 = Prime
2^3 + 1 = 9 = Not Prime
Insufficient

(2) p is an even number.
2^2 + 1 = 5 = Prime
2^4 + 1 = 17 = Prime
2^6 + 1 = 65 = Not Prime
Insufficient

Combining both :
We only know of one number that is both Even & Prime i.e, 2
2^2 + 1 = 5 = Prime
Sufficient

Hence, C.
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If p is a positive integer, is 2^p + 1 a prime number?  [#permalink]

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04 Aug 2018, 13:00
Bunuel wrote:
If p is a positive integer, is 2^p + 1 a prime number?

(1) p is a prime number.
(2) p is an even number.

NEW question from GMAT® Quantitative Review 2019

(DS19208)

Given p is a positive integer.
Question stem:- Is $$2^p + 1$$ a prime number?

St1:-p is a prime number
We don't have sufficient info to determine whether $$2^p + 1$$ is a prime number or not.
If p=2, then $$2^p+1=5$$,which is prime.
If p=3, then $$2^p + 1=9$$, which is not prime
Insufficient.

St2:- p is an even number
If p=2 then $$2^p + 1=5$$, which is prime.
If p=0, then $$2^p + 1 =2^0+1=1+1=2$$, which is prime
If p=6, then $$2^p + 1 =2^6+1=64+1=65$$, which is not prime
Insufficient.

Combining, p has to be both even and prime number; we know p=2 is the only even prime number.
Now, $$2^p+1=2^2+1=4+1=5$$, which is prime.
Sufficient.

Ans. (C)
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If p is a positive integer, is 2^p + 1 a prime number?  [#permalink]

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04 Aug 2018, 13:52
Bunuel wrote:
If p is a positive integer, is 2^p + 1 a prime number?

(1) p is a prime number.
(2) p is an even number.

NEW question from GMAT® Quantitative Review 2019

(DS19208)

Statement 1: Test the data. plug value for p. NOT sufficient.

Statement 2: This one is bit tricky. For even integer 0, 2 , 4 we get prime. But if we stop after that we surely miss something . even integer 6 changes the scenario. NOT sufficient.

Combining both options:

Both option together limited our choice. Only 2 is valid here. 2 is even and prime. Sufficient .

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Re: If p is a positive integer, is 2^p + 1 a prime number?  [#permalink]

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05 Aug 2018, 05:32
Bunuel wrote:
If p is a positive integer, is 2^p + 1 a prime number?

(1) p is a prime number.
(2) p is an even number.

NEW question from GMAT® Quantitative Review 2019

(DS19208)

my logic:
a. 2^p + 1 = prime for only for p= 1,2,4 etc.. non prime for p=3.. 2 answers.. Yes/no -->NS
b. p = even.. this can be checked using p =2 (yes) p=3 (no)

combined .. p = even prime (only ) 2.

2^p + 1 = not prime
C
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Re: If p is a positive integer, is 2^p + 1 a prime number?  [#permalink]

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05 Aug 2018, 06:25
Here is my method for DS, which I call pre-workout. In most of the times, when I see formulae in question stem of DS I try first to modify them.

2^p + 1 a prime number? We know that any prime number can be expressed as 6n+1 or 6n-1, so

a. 2^p + 1=6n+1, 2^p =6n, 2^p =2*3*n - never a case, because 2^p can not have 3.
b. 2^p + 1=6n-1, 2^p =6n-2, 2^p =2*(3n-1), (3n-1) - has to be 2, 8, 32 ..., or p=2,4,6... - we have to find an option that gives us p=even

(1)+(2) states that p=2 - the only prime number that is even. Answer (C)

P.S. From the beginning this method seems to be slower, but it certanly has several benefits:

1. It allows you to narrow down to information that you are looking for, and not to be confused with fluff in options
2. In some cases it's easier to conduct calculations with only one formula in the beginning
3. Sometimes plug-in method can be very messy, because you have a lot of scenarios.
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Re: If p is a positive integer, is 2^p + 1 a prime number?  [#permalink]

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05 Aug 2018, 12:18
Bunuel wrote:
If p is a positive integer, is 2^p + 1 a prime number?

(1) p is a prime number.
(2) p is an even number.

NEW question from GMAT® Quantitative Review 2019

(DS19208)

Instead of solving -- (2^p)+ 1, I started solving for 2^ (p+1).
How to distinguish between the two if the question is written the way its mentioned above.
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If p is a positive integer, is 2^p + 1 a prime number?  [#permalink]

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Updated on: 05 Aug 2018, 12:57
1
tatz wrote:
Instead of solving -- (2^p)+ 1, I started solving for 2^ (p+1).
How to distinguish between the two if the question is written the way its mentioned above.

2^p + 1
If it was to mean 2^(p + 1), there would always be a bracket for p+1.
That is why they say GMAT quant is not much tough, but in hurry and pressure you might misinterpret the nuances.
So, be careful and always be on the watch out for such confusions.
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Originally posted by sudarshan22 on 05 Aug 2018, 12:27.
Last edited by sudarshan22 on 05 Aug 2018, 12:57, edited 1 time in total.
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Re: If p is a positive integer, is 2^p + 1 a prime number?  [#permalink]

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05 Aug 2018, 12:54
sudarshan22 wrote:
tatz wrote:
Instead of solving -- (2^p)+ 1, I started solving for 2^ (p+1).
How to distinguish between the two if the question is written the way its mentioned above.

2^p + 1
If it was to mean 2^(p + 1), there would always be a bracket for p+1.
That is why they say GMAT quant is not much touch, but when in hurry and pressure you might misinterpret the nuances.
So, be careful and always be on the watch out for such confusions.

Thanks Sudarshan.. it means we should not complicate the statement. Solve what is appears to be at first sight. But literally, writing everything in same line may confuse people...
Re: If p is a positive integer, is 2^p + 1 a prime number? &nbs [#permalink] 05 Aug 2018, 12:54
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