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exponent equation

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Manager
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Joined: 26 Dec 2008
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Schools: Booth (Admit R1), Sloan (Ding R1), Tuck (R1)
exponent equation [#permalink]

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New post 06 Mar 2009, 10:21
This topic is locked. If you want to discuss this question please re-post it in the respective forum.

Guys, just for fun.. this question used to be a popular hazing question in my college. I have adapted it to PS format.

===

if x and y are integers such that x not equal to y, atleast how many solutions exist for following equation:

x^y + y^x = n such that n is an integer variable belonging to [1000, 1099]

A) 100
B) 200
C) 250
D) 350
E) 450

Hope there are members from my college who would recognize this one :)

Kudos [?]: 14 [0], given: 0

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Joined: 29 Aug 2007
Posts: 2472

Kudos [?]: 843 [0], given: 19

Re: exponent equation [#permalink]

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New post 06 Mar 2009, 14:05
xyz21 wrote:
Guys, just for fun.. this question used to be a popular hazing question in my college. I have adapted it to PS format.

===

if x and y are integers such that x not equal to y, at least how many solutions exist for following equation:

x^y + y^x = n such that n is an integer variable belonging to [1000, 1099]

A) 100
B) 200
C) 250
D) 350
E) 450

Hope there are members from my college who would recognize this one :)


Since x and y are integers and x is not equal to y, the only values for x and y that satisfy the given equation is: either x = 1 and y = 1000 or x = 1000 and y = 1.

If one value (x/y) is 2, the other value (y/x) ranges from 9.82 to 9.97.
Similarly, if one value is 2, the other value ranges arround 6.10.

Are the answer choices correct? :roll:
_________________

Verbal: http://gmatclub.com/forum/new-to-the-verbal-forum-please-read-this-first-77546.html
Math: http://gmatclub.com/forum/new-to-the-math-forum-please-read-this-first-77764.html
Gmat: http://gmatclub.com/forum/everything-you-need-to-prepare-for-the-gmat-revised-77983.html


GT

Kudos [?]: 843 [0], given: 19

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Re: exponent equation [#permalink]

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New post 06 Mar 2009, 22:41
xyz21 wrote:
Guys, just for fun.. this question used to be a popular hazing question in my college. I have adapted it to PS format.

===

if x and y are integers such that x not equal to y, atleast how many solutions exist for following equation:

x^y + y^x = n such that n is an integer variable belonging to [1000, 1099]

A) 100
B) 200
C) 250
D) 350
E) 450

Hope there are members from my college who would recognize this one :)


Of course, with those answer choices, A must be correct (if this is a GMAT question). Even if you find there are 940 solutions, then certainly there are 'at least 100'.

But as GMATTiger pointed out above, we have the solution x = 1, y = 1000. We actually have solutions x = 1, and y = 999, 1000, 1001, 1002, ... 1098, giving us a hundred solutions. The equation is symmetric in x and y, so if you consider the solutions y = 1, x = 999, 1000, 1001... 1098 to be different from the first set of solutions, we then get up to 200. If my mental arithmetic was right, there aren't any other solutions, though I checked that very quickly.
_________________

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If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

Kudos [?]: 1954 [0], given: 6

Manager
Manager
User avatar
Joined: 26 Dec 2008
Posts: 58

Kudos [?]: 14 [0], given: 0

Schools: Booth (Admit R1), Sloan (Ding R1), Tuck (R1)
Re: exponent equation [#permalink]

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New post 07 Mar 2009, 06:23
IanStewart wrote:
xyz21 wrote:
Guys, just for fun.. this question used to be a popular hazing question in my college. I have adapted it to PS format.

===

if x and y are integers such that x not equal to y, atleast how many solutions exist for following equation:

x^y + y^x = n such that n is an integer variable belonging to [1000, 1099]

A) 100
B) 200
C) 250
D) 350
E) 450

Hope there are members from my college who would recognize this one :)


Of course, with those answer choices, A must be correct (if this is a GMAT question). Even if you find there are 940 solutions, then certainly there are 'at least 100'.

But as GMATTiger pointed out above, we have the solution x = 1, y = 1000. We actually have solutions x = 1, and y = 999, 1000, 1001, 1002, ... 1098, giving us a hundred solutions. The equation is symmetric in x and y, so if you consider the solutions y = 1, x = 999, 1000, 1001... 1098 to be different from the first set of solutions, we then get up to 200. If my mental arithmetic was right, there aren't any other solutions, though I checked that very quickly.


Bingo.

Thanks, Ian -- I should have framed the wording as "..how many solutions.." and should have avoided use of at least. You got it, those 200 form the solution set. Good show!

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Re: exponent equation [#permalink]

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New post 07 Mar 2009, 06:45
IanStewart wrote:
xyz21 wrote:
Guys, just for fun.. this question used to be a popular hazing question in my college. I have adapted it to PS format.

===

if x and y are integers such that x not equal to y, atleast how many solutions exist for following equation:

x^y + y^x = n such that n is an integer variable belonging to [1000, 1099]

A) 100
B) 200
C) 250
D) 350
E) 450

Hope there are members from my college who would recognize this one :)


Of course, with those answer choices, A must be correct (if this is a GMAT question). Even if you find there are 940 solutions, then certainly there are 'at least 100'.

But as GMATTiger pointed out above, we have the solution x = 1, y = 1000. We actually have solutions x = 1, and y = 999, 1000, 1001, 1002, ... 1098, giving us a hundred solutions. The equation is symmetric in x and y, so if you consider the solutions y = 1, x = 999, 1000, 1001... 1098 to be different from the first set of solutions, we then get up to 200. If my mental arithmetic was right, there aren't any other solutions, though I checked that very quickly.


Thanks Ian, you got it.

I knew that I was missing something. I pondered for a while but missed that one.....
I went for 1, 2, 3, 4, 5,.............. but not for 1001, 1002...... and 1099.
_________________

Verbal: http://gmatclub.com/forum/new-to-the-verbal-forum-please-read-this-first-77546.html
Math: http://gmatclub.com/forum/new-to-the-math-forum-please-read-this-first-77764.html
Gmat: http://gmatclub.com/forum/everything-you-need-to-prepare-for-the-gmat-revised-77983.html


GT

Kudos [?]: 843 [0], given: 19

Re: exponent equation   [#permalink] 07 Mar 2009, 06:45
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