Brent@GMATPrepNow wrote:If x and y are non-zero integers, and (-x)^x = y^y, what is the value of x?
(1) x + y = 0
(2) x is odd
Target question: What is the value of x?
Given: x and y are non-zero integers, and [m](-x)^x = y^y[/m]
When we see this given information, we should be thinking of a few different ways for the equation to hold true.
For example, if x = 1, then (-x)^x = (-1)^1 = -1. What value of y is necessary for y^y to equal -1? Well, if y = -1, then y^y = (-1)^(-1) = 1/(-1) = -1
So, one possible solution to the given equation is
x = 1 and y = -1
Using very similar logic, we can see that
x = -1 and y = 1 is another possible solution to the given equation
Now let's check the statements....
Statement 1: x + y = 0
As we can see from our earlier work, there are at least two possible solutions that satisfy statement 1:
Case a:
x = 1 and y = -1. In this case, the answer to the target question is
x = 1
Case b:
x = -1 and y = 1. In this case, the answer to the target question is
x = -1
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: x is odd
We should recognize that we can RE-USE the same values we used to show that statement 1 is not sufficient (since 1 and -1 are both ODD)
That is....
Case a:
x = 1 and y = -1. In this case, the answer to the target question is
x = 1
Case b:
x = -1 and y = 1. In this case, the answer to the target question is
x = -1
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the
same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words:
Case a:
x = 1 and y = -1. In this case, the answer to the target question is
x = 1
Case b:
x = -1 and y = 1. In this case, the answer to the target question is
x = -1
Since we cannot answer the
target question with certainty, the combined statements are NOT SUFFICIENT
Answer: E
Cheers,
Brent
