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# exponent equation

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Manager
Joined: 26 Dec 2008
Posts: 58

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Schools: Booth (Admit R1), Sloan (Ding R1), Tuck (R1)

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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

SVP
Joined: 29 Aug 2007
Posts: 2472

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

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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.

_________________

Gmat: http://gmatclub.com/forum/everything-you-need-to-prepare-for-the-gmat-revised-77983.html

GT

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

GMAT Tutor
Joined: 24 Jun 2008
Posts: 1339

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

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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.
_________________

GMAT Tutor in Toronto

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
Joined: 26 Dec 2008
Posts: 58

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

Schools: Booth (Admit R1), Sloan (Ding R1), Tuck (R1)

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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!

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

SVP
Joined: 29 Aug 2007
Posts: 2472

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

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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.
_________________

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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