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If x and y are distinct positive integers, what is the value

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If x and y are distinct positive integers, what is the value [#permalink] New post 18 Jul 2010, 07:11
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If x and y are distinct positive integers, what is the value of x^4 - y^4?

(1) (y^2 + x^2)(y + x)(x - y) = 240
(2) x^y = y^x and x > y
[Reveal] Spoiler: OA
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Re: 4 Powerful XYs [#permalink] New post 18 Jul 2010, 07:28
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ankitmania wrote:
If x and y are distinct positive integers, what is the value of x^4 - y^4?

1. (y^2 + x^2)(y + x)(x - y) = 240
2. x^y = y^x and x > y


Important property to know: x^2-y^2=(x+y)(x-y).

Given: x and y are distinct positive integers. Question: x^4-y^4=?


(1) (y^2+x^2)(y+x)(x-y)=240 --> (y^2+x^2)(x^2-y^2)=240 --> x^4-y^4=240. Sufficient.

(2) x^y = y^x and x>y, also x and y are distinct positive integers --> only one such pair is possible x=4>y=2: x^y=4^2=16=2^4=y^x --> x^4-y^4=240. Sufficient.

Answer: D.
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Re: 4 Powerful XYs [#permalink] New post 18 Jul 2010, 08:33
Bunuel wrote:
only one such pair is possible x=4>y=2: x^y=4^2=16=2^4=y^x --> x^4-y^4=240. Sufficient.


How do you come to this conclusion? Just picking numbers/knowing or is there some mathematical rule for this? I understand that it revolves around the power of 2, but can't put my finger on it.
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Re: 4 Powerful XYs [#permalink] New post 18 Jul 2010, 10:05
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Suvorov wrote:
Bunuel wrote:
only one such pair is possible x=4>y=2: x^y=4^2=16=2^4=y^x --> x^4-y^4=240. Sufficient.


How do you come to this conclusion? Just picking numbers/knowing or is there some mathematical rule for this? I understand that it revolves around the power of 2, but can't put my finger on it.


I think it's worth remembering that 4^2=16=2^4, I've seen several GMAT questions on number properties using this (another useful property 8^2=4^3=2^6=64).

But if you don't know this property:

Given: x^y = y^x and x>y, also x and y are distinct positive integers.
Couple of things:
x and y must be either distinct positive odd integers or distinct positive even integers (as odd in ANY positive integer power is odd and even in ANY positive integer power is even).

After testing several options you'll see that x=4 and y=2 is the only possible scenario: because, when y\geq{2} and x>{4}, then x^y (bigger value in smaller power) will be always less than y^x (smaller value in bigger power): 5^3<3^5, or 6^2<2^6, or 8^2<2^8, or 10^2<2^{10}, ....
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Re: If x and y are distinct positive integers, what is the value [#permalink] New post 15 Oct 2013, 05:20
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Re: If x and y are distinct positive integers, what is the value   [#permalink] 15 Oct 2013, 05:20
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