|
Author |
Message |
|
TAGS:
|
|
|
Manager
Joined: 09 May 2009
Posts: 213
Followers: 1
Kudos [?]:
45
[0], given: 13
|
Question Stats:
0% (00:00) correct
100% (00:47) wrong based on 1 sessions
Find the number of pairs of positive integers (x, y) such that x^6 = y^2 + 127. (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 pls share if u used a quick approach to solve this
_________________
GMAT is not a game for losers , and the moment u decide to appear for it u are no more a loser........ITS A BRAIN GAME
|
|
|
|
|
|
|
CEO
Joined: 29 Aug 2007
Posts: 2530
Followers: 41
Kudos [?]:
357
[0], given: 19
|
Re: x^6 = y^2 + 127 [#permalink]
 12 Dec 2009, 00:06
xcusemeplz2009 wrote: Find the number of pairs of positive integers (x, y) such that x^6 = y^2 + 127.
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
pls share if u used a quick approach to solve this First note that: x and y are +ve integers i.e. x and y cannot be -ves, zeros and fractions. Given that:- x^6 = y^2 + 127 x can have a value of 1....................n, however x>2 because: If x = 1, x^6 = 1 and y = sqrt(1-127). y is an irrational number. Not possible. If x = 2, x^6 = 64 and y = sqrt(64-127). y is an irrational number. Not possible. If x = 3, x^6 = 729 and y = sqrt(729-127) = sqrt(602). Now y is a fraction. Not possible. If x = 4, x^6 = 64x64 and y = sqrt{(64x64) -127} = .......? Now its not possible to do any calculation beyond this point in 2-5 minuets. Until and unless there is a quick approach, I would say this not a gmat-type question. I would love to see if anybody has a quick approach. I thought its a good question however it turned to be a tough one.
_________________
Verbal: new-to-the-verbal-forum-please-read-this-first-77546.html
Math: new-to-the-math-forum-please-read-this-first-77764.html
Gmat: everything-you-need-to-prepare-for-the-gmat-revised-77983.html
GT
|
|
|
|
|
|
Manager
Joined: 12 Oct 2008
Posts: 59
Followers: 1
Kudos [?]:
1
[0], given: 3
|
Re: x^6 = y^2 + 127 [#permalink]
12 Dec 2009, 01:48
Thats correct, even I used brute force. But can we find some better approach for similar question, where constant may be changed i.e. 64 or exponent may be changed. ?
|
|
|
|
|
|
Senior Manager
Joined: 02 Aug 2009
Posts: 274
Followers: 3
Kudos [?]:
59
[1] , given: 1
|
Re: x^6 = y^2 + 127 [#permalink]
12 Dec 2009, 07:16
1
This post received
KUDOS
HI.. a quick appch i can think of..
the eq can be written as... x^6-y^2=127...x^6-y^2=(x^3-y)(x^3+y)....which means 127 is product of two int... but if u look at 127..it is prime number... so there is no pair of positive integers which satisfies the condition ...ANS-0
|
|
|
|
|
|
GMAT Instructor
Joined: 24 Jun 2008
Posts: 973
Location: Toronto
Followers: 167
Kudos [?]:
443
[1] , given: 3
|
Re: x^6 = y^2 + 127 [#permalink]
12 Dec 2009, 08:03
1
This post received
KUDOS
xcusemeplz2009 wrote: Find the number of pairs of positive integers (x, y) such that x^6 = y^2 + 127.
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
pls share if u used a quick approach to solve this chetan2u's approach above is excellent, though there's one problem with the analysis. We have (x^3 + y)(x^3 - y) = 127 so x^3 + y and x^3 - y must be factors of 127. Since 127 is prime, the only possibility is that x^3 + y = 127 and x^3 - y = 1. Now, x cannot be greater than 5, since that would make x^3 larger than 127, and since x^3 must be greater than y, x could only be 4 or 5. Still testing these values, we do find that x = 4 and y = 63 gives a legitimate solution here, so there is one pair of values that works.
_________________
Nov 2011: After years of development, I am now making my advanced Quant books and high-level problem sets available for sale. Contact me at ianstewartgmat at gmail.com for details.
Private GMAT Tutor based in Toronto
|
|
|
|
|
|
Manager
Joined: 13 Aug 2009
Posts: 208
Followers: 3
Kudos [?]:
60
[0], given: 16
|
Re: x^6 = y^2 + 127 [#permalink]
12 Dec 2009, 08:06
chetan2u wrote: HI.. a quick appch i can think of..
the eq can be written as... x^6-y^2=127...x^6-y^2=(x^3-y)(x^3+y)....which means 127 is product of two int... but if u look at 127..it is prime number... so there is no pair of positive integers which satisfies the condition ...ANS-0 This is definitely a tough one... but I do not think the answer is 0. chetan2u had a great approach, same as the one I used but since 127 is a prime number I went a step further: (x^3-y)(x^3+y)=127 (x^3-y)(x^3+y)=(1)(127) From here I plugged in different values of x and y to see if I can get (x^3-y) to equal 1 and (x^3+y) to equal 127. Because we know that (x^3-y) must equal 1, y must be x^3-1. x=1, not possible x=2, not possible x=3, not possible x=4, y=63 (x^3-y)(x^3+y)=127 (4^3-63)*(4^3+63)=127 x=5, not possible So answer is B. 1 Can someone please verify the OA?
|
|
|
|
|
|
Senior Manager
Joined: 02 Aug 2009
Posts: 274
Followers: 3
Kudos [?]:
59
[0], given: 1
|
Re: x^6 = y^2 + 127 [#permalink]
12 Dec 2009, 08:25
hi ianstewart...
thanks a lot... should not have skipped my mind but it seems in a hurry, just overlooked it...
lesson learnt-give a thought before proceeding to next step..
|
|
|
|
|
|
CEO
Joined: 29 Aug 2007
Posts: 2530
Followers: 41
Kudos [?]:
357
[0], given: 19
|
Re: x^6 = y^2 + 127 [#permalink]
12 Dec 2009, 13:10
chetan2u wrote: HI.. a quick appch i can think of..
the eq can be written as... x^6-y^2=127...x^6-y^2=(x^3-y)(x^3+y)....which means 127 is product of two int... but if u look at 127..it is prime number... so there is no pair of positive integers which satisfies the condition ...ANS-0 Thats a good point. x^6-y^2 = 127 (x^3-y)(x^3+y) = 127 x 1 Since x and y both are +ves, (x^3-y) = 1 and (x^3+y) = 127 As I mentioned earlier, x >2. Now the possibilities for x are <5. So x could be 3 or 4 or 5. After few trails, x = 4. Agree with OA as B.
_________________
Verbal: new-to-the-verbal-forum-please-read-this-first-77546.html
Math: new-to-the-math-forum-please-read-this-first-77764.html
Gmat: everything-you-need-to-prepare-for-the-gmat-revised-77983.html
GT
|
|
|
|
|
|
Senior Manager
Joined: 21 Jul 2009
Posts: 371
Schools: LBS, INSEAD, IMD, ISB - Anything with just 1 yr program.
Followers: 11
Kudos [?]:
74
[0], given: 22
|
Re: x^6 = y^2 + 127 [#permalink]
12 Dec 2009, 15:21
I believe, solving these kinds of questions is key to scoring well on the test. When we never know which question is experimental, a tough one as worse as this can always waste time and lower scores. I have had similar questions and I did not have sufficient practise facing them and lost valuable time. The key is to not panic and try to think at least one step beyond or differently for every 3 to 5 seconds when facing a tough problem.
_________________
I am AWESOME and it's gonna be LEGENDARY!!!
|
|
|
|
|
|
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 3104
Location: Pune, India
Followers: 567
Kudos [?]:
1994
[2] , given: 92
|
Re: x^6 = y^2 + 127 [#permalink]
21 Oct 2011, 03:34
2
This post received
KUDOS
xcusemeplz2009 wrote: Find the number of pairs of positive integers (x, y) such that x^6 = y^2 + 127.
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
pls share if u used a quick approach to solve this Here is what I would do: First thing that comes to my mind is that 127 is prime. Well then, I can't do anything with it right now. Then I see that there are some squares (x^3)^2 - y^2 = 127Let's say x^3 = aa^2 - y^2 = 127(a+y)(a-y) = 127*1We know that a and y are both positive integers. Therefore, their sum, a+y = 127 and their difference, a-y = 1. It is obvious that a and y must be 64 and 63. (or you can solve the two equations simultaneously to get the values for a and y) If a = 64 = x^3, x must be 4. So there is only one pair of values (4, 63). The question is straight forward because 127 is prime. You get only one pair of values. If instead, we have a composite number with many factors, we need to find the possible values of a and y and then see which values of a work for us.
_________________
Karishma
Veritas Prep | GMAT Instructor
My Blog
Save 10% on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.
Veritas Prep Reviews
|
|
|
|
|
|
|
Re: x^6 = y^2 + 127
[#permalink]
21 Oct 2011, 03:34
|
|
|
|
|
|
|
|
|
|
|
close
