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

Question

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

Bunuel · Answer

Similar question: if-x-and-y-are-distinct-positive-integers-142005.html

arichinna · Answer

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.

Bunuel · Answer

Notice that 4^16 ≠ 16^4.

NatC · Answer

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.

Nunuboy1994 · Answer

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"

gvij2017 · Answer

If I am not mistaken, 0 and 1 does't satisfy statement 2. 
0^1 not equals to 1^0

bumpbot · Answer

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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 ....

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