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

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Intern
Joined: 24 Feb 2015
Posts: 45
If x and y are distinct positive integers, what is the value of ....  [#permalink]

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10 Sep 2015, 15:00
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Difficulty:

95% (hard)

Question Stats:

40% (02:12) correct 60% (02:18) wrong based on 203 sessions

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If x and y are distinct positive integers, what is the value of $$x^4 - y^4$$?

1. $$(y^2 + x^2)(y + x)(x - y) > 100$$
2. $$x^y = y^x$$
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Joined: 02 Sep 2009
Posts: 58396
Re: If x and y are distinct positive integers, what is the value of ....  [#permalink]

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10 Sep 2015, 22:04
1
1
Prajat wrote:
If x and y are distinct positive integers, what is the value of $$x^4 - y^4$$?

1. $$(y^2 + x^2)(y + x)(x - y) > 100$$
2. $$x^y = y^x$$

Similar question: if-x-and-y-are-distinct-positive-integers-142005.html
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Joined: 24 Jun 2013
Posts: 32
Re: If x and y are distinct positive integers, what is the value of ....  [#permalink]

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13 Mar 2017, 00:51
Prajat wrote:
If x and y are distinct positive integers, what is the value of $$x^4 - y^4$$?

1. $$(y^2 + x^2)(y + x)(x - y) > 100$$
2. $$x^y = y^x$$

Hi,

I could infer the answer to be "E".

1. $$(y^2 + x^2)(y + x)(x - y) > 100$$

By solving, $$(x^4 - y^4) > 100$$. The Value cannot not be decided as the set is infinite (> 100). Hence it is insufficient.

2. $$x^y = y^x$$

x and y being two distinct positive integers, the values can take one of the forms as below,
(a) y=2, x=4
(b) y=4, x=16 etc.
Hence insufficient.

By combining 1 and 2, many values exist,

(a) y=2, x=4, then $$(x^4 - y^4) > 100$$ becomes 240, which is >100
(b) y=4, x=16, then $$(x^4 - y^4) > 100$$ becomes 65280, which is also >100.

so no one value can be inferred by combining 1 and 2.

Please let me know any other alternative views.

Thanks.
Math Expert
Joined: 02 Sep 2009
Posts: 58396
Re: If x and y are distinct positive integers, what is the value of ....  [#permalink]

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13 Mar 2017, 01:01
arichinna wrote:
Prajat wrote:
If x and y are distinct positive integers, what is the value of $$x^4 - y^4$$?

1. $$(y^2 + x^2)(y + x)(x - y) > 100$$
2. $$x^y = y^x$$

Hi,

I could infer the answer to be "E".

1. $$(y^2 + x^2)(y + x)(x - y) > 100$$

By solving, $$(x^4 - y^4) > 100$$. The Value cannot not be decided as the set is infinite (> 100). Hence it is insufficient.

2. $$x^y = y^x$$

x and y being two distinct positive integers, the values can take one of the forms as below,
(a) y=2, x=4
(b) y=4, x=16etc.
Hence insufficient.

By combining 1 and 2, many values exist,

(a) y=2, x=4, then $$(x^4 - y^4) > 100$$ becomes 240, which is >100
(b) y=4, x=16, then $$(x^4 - y^4) > 100$$ becomes 65280, which is also >100.

so no one value can be inferred by combining 1 and 2.

Please let me know any other alternative views.

Thanks.

Notice that 4^16 ≠ 16^4.
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Joined: 11 Jun 2016
Posts: 4
Re: If x and y are distinct positive integers, what is the value of ....  [#permalink]

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16 Mar 2017, 08:33
1
Good question!

While B might look like the answer (I spent a good amount of time trying to understand why it wasn't), C is correct.

Statement 1 gives the lowest limit of the, i.e. x^4-y^4 >100. Basically, the result will be a positive number greater than 100.
Statement B defines a relationship between x and y such that only the numbers 2 and 4 satisfy this relationship. Butwe either X or Y could be 2 or 4. Thus, we do not know the value for each variable. If x = 2 and y = 4, then we get a negative result, but if x is 4 and y is 2, then we get a postive result that is quite large

Combining both statements, we can rule out x=2 and y=4, as we know that the result must be positive and greater than 100.
This gives the correct answer, x=4 and y=2.
Hence C.
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Re: If x and y are distinct positive integers, what is the value of ....  [#permalink]

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29 May 2017, 20:06
2
Prajat wrote:
If x and y are distinct positive integers, what is the value of $$x^4 - y^4$$?

1. $$(y^2 + x^2)(y + x)(x - y) > 100$$
2. $$x^y = y^x$$

They key words in this problem are "distinct positive integers."

Statement 1

(y^2 + x^2)(y + x)(x - y) > 10
(y^2 + x^2) (xy- y^2 + x^2 -yx)
xy^3- y^4 + x^2y^2 - y^3x + x^3y - x^2y^2 + x^4 -yx^3 ( notice terms that cancel)
-y^4 + x^4
x^4-y^4 >100

Insufficient because there are infinite variables that can satisfy this inequality

Statement 2

x^y=y^x

only 0,1 or 2 and 4 can satisfy this equation; however, the integers must be both positive and distinct. Therefore, the set of integers must be 2 and 4- but x and y cannot be distinguished- x could be 2 or x could be 4

Statement 1 and 2

Using both statements it can be inferred that x must be 4 because x must be greater than 100.

Hence
"C"
Director
Joined: 09 Aug 2017
Posts: 500
Re: If x and y are distinct positive integers, what is the value of ....  [#permalink]

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30 Oct 2018, 20:07
If I am not mistaken, 0 and 1 does't satisfy statement 2.
0^1 not equals to 1^0

Nunuboy1994 wrote:
Prajat wrote:
If x and y are distinct positive integers, what is the value of $$x^4 - y^4$$?

1. $$(y^2 + x^2)(y + x)(x - y) > 100$$
2. $$x^y = y^x$$

They key words in this problem are "distinct positive integers."

Statement 1

(y^2 + x^2)(y + x)(x - y) > 10
(y^2 + x^2) (xy- y^2 + x^2 -yx)
xy^3- y^4 + x^2y^2 - y^3x + x^3y - x^2y^2 + x^4 -yx^3 ( notice terms that cancel)
-y^4 + x^4
x^4-y^4 >100

Insufficient because there are infinite variables that can satisfy this inequality

Statement 2

x^y=y^x

only 0,1 or 2 and 4 can satisfy this equation; however, the integers must be both positive and distinct. Therefore, the set of integers must be 2 and 4- but x and y cannot be distinguished- x could be 2 or x could be 4

Statement 1 and 2

Using both statements it can be inferred that x must be 4 because x must be greater than 100.

Hence
"C"
Re: If x and y are distinct positive integers, what is the value of ....   [#permalink] 30 Oct 2018, 20:07
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