Prime Number

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rohu27: Legendary Member; Posts: 586; Joined: Tue Jan 19, 2010 4:38 amFollowed by: 5 members; Thanked: 31 times; GMAT Score:730

by rohu27 » Thu Apr 07, 2011 9:26 pm

Source:gmatprepnow.com

If K is a positive integer, is (2^k)-1 a prime number?

1)K is a prime number
2)K has exactly 2 positive divisors

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Source: Beat The GMAT — Data Sufficiency | Thu Apr 07, 2011 9:26 pm

vineeshp: Legendary Member; Posts: 965; Joined: Thu Jan 28, 2010 12:52 amFollowed by: 34 members; Thanked: 156 times; GMAT Score:720

by vineeshp » Thu Apr 07, 2011 10:04 pm

D Either statement is sufficient.

I dont see a mathematical way of solving this, Maybe experts have. But 2 ^ 1 -1, 2 ^ 3 -1 , 2 ^ 5 -1, 2 ^ 7 -1, 2 ^ 11 -1 all lead us to prime numbers. I think we can safely conclude that they lead us to prime numbers.

1) k is prime, so 2 ^ k -1 leads us to prime.
2) Has stated the same thing in another way. 2 divisors = Prime.

Vineesh,
Just telling you what I know and think. I am not the expert.

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HSPA: Legendary Member; Posts: 1101; Joined: Fri Jan 28, 2011 7:26 amFollowed by: 13 members; Thanked: 47 times; GMAT Score:640

by HSPA » Fri Apr 08, 2011 1:20 am

+1 for D.. easy way is only substitution I believe

First take: 640 (50M, 27V) - RC needs 300% improvement
Second take: coming soon..
Regards,
HSPA.

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Fri Apr 08, 2011 6:01 am

rohu27 wrote:Source:gmatprepnow.com

If K is a positive integer, is (2^k)-1 a prime number?

1)K is a prime number
2)K has exactly 2 positive divisors

Hi there!

First of all, statements (1) and (2) are equivalent (in fact the same), therefore the answer to the question is certainly (D) or (E).

The correct answer is (E), as shown below:

> Take k=2 , then 2^k-1 = 3 answering in the affirmative;
> Take k=11, then 2^k-1 = 2047 = 23*29 answering in the negative.

For the interested readers in this very famous subject ("Mersenne primes") I suggest:

https://primes.utm.edu/mersenne/index.html

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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ldoolitt: Master | Next Rank: 500 Posts; Posts: 184; Joined: Sat Apr 14, 2007 9:23 am; Location: Madison, WI; Thanked: 17 times

by ldoolitt » Fri Apr 08, 2011 6:30 am

See above for answer.

If the question had asked the reverse, ie "K is a positive integer, 2^k-1 is prime, is k prime?" then you could answer that without any statements as yes.

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Fri Apr 08, 2011 6:45 am

rohu27 wrote:Source:gmatprepnow.com

If K is a positive integer, is (2^k)-1 a prime number?

1)K is a prime number
2)K has exactly 2 positive divisors

This is based on a famous problem in Number Theory, as fskilnik points out, but it most certainly is not a realistic GMAT question. To see why Statement 1 is not sufficient, you need to plug in a value for k at least as large as 11, and there is no quick way to prime factorize 2^11 - 1. The GMAT will never ask you to do that.

There is no formula for generating prime numbers, and I suppose if you know that, you can answer E here instantly, but that's not something the GMAT will ever test.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com

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rohu27: Legendary Member; Posts: 586; Joined: Tue Jan 19, 2010 4:38 amFollowed by: 5 members; Thanked: 31 times; GMAT Score:730

by rohu27 » Fri Apr 08, 2011 7:34 am

Thanks Ian and Fabio.
The problem involves huge calculations, just wnated to make sure if there was other way round, but Ian's reply soothes me.

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force5: Legendary Member; Posts: 582; Joined: Tue Mar 08, 2011 12:48 amFollowed by: 6 members; Thanked: 61 times; GMAT Score:740

by force5 » Mon Apr 11, 2011 12:15 pm

i bet it soothes many more...

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Apr 11, 2011 1:05 pm

As the one who created that question, I should point out that it was never intended to be an actual practice question. As Ian pointed out, the question is beyond the scope of what the GMAT tests.

The question is actually from a Data Sufficiency lesson on Guessing Strategically (https://youtu.be/2GOt0yZbrVg). In this video we examine how we might eliminate answer choices if we cannot answer a certain DS question. So, even though the question is well beyond the scope of the GMAT, we can easily eliminate answer choices A, B and C, since statements 1 and 2 provide the same information, and when the two statements provide the same information, the correct answer must be either D or E

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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rohu27: Legendary Member; Posts: 586; Joined: Tue Jan 19, 2010 4:38 amFollowed by: 5 members; Thanked: 31 times; GMAT Score:730

by rohu27 » Mon Apr 11, 2011 6:55 pm

Brent Hanneson wrote:As the one who created that question, I should point out that it was never intended to be an actual practice question. As Ian pointed out, the question is beyond the scope of what the GMAT tests.

The question is actually from a Data Sufficiency lesson on Guessing Strategically (https://youtu.be/2GOt0yZbrVg). In this video we examine how we might eliminate answer choices if we cannot answer a certain DS question. So, even though the question is well beyond the scope of the GMAT, we can easily eliminate answer choices A, B and C, since statements 1 and 2 provide the same information, and when the two statements provide the same information, the correct answer must be either D or E

Cheers,
Brent

Thanks Brent. I did pick it up from tht video and was scared when i saw the solution (calculate 2^11).But anyways i get to learn a new quant concept though may not be useful for gmat.
the video was really good btw. things you already knw but tend to ignore at pressure times.

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