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

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

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New post 10 Sep 2015, 14:00
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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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Re: If x and y are distinct positive integers, what is the value of ....  [#permalink]

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New post 10 Sep 2015, 21:04
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Re: If x and y are distinct positive integers, what is the value of ....  [#permalink]

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New post 12 Mar 2017, 23:51
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.

Hence, the answer is E.

Please let me know any other alternative views.

Thanks.
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Re: If x and y are distinct positive integers, what is the value of ....  [#permalink]

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New post 13 Mar 2017, 00:01
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.

Hence, the answer is E.

Please let me know any other alternative views.

Thanks.


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

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New post 16 Mar 2017, 07:33
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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New post 29 May 2017, 19:06
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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"
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Re: If x and y are distinct positive integers, what is the value of ....  [#permalink]

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New post 30 Oct 2018, 19:07
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"
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Re: If x and y are distinct positive integers, what is the value of .... &nbs [#permalink] 30 Oct 2018, 19:07
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