What is the value of positive integer x?

Hi sterling19,

While you've selected the correct answer, your logic is incomplete/incorrect.

We're told that X is a POSITIVE INTEGER.

Fact 1: X^W = 1

You correctly deduced that W could be 0 or 1, but W can actually be ANY VALUE (if X = 1, 1^(any) = 1). In that same way, X can be ANY positive integer (if X = 0, (any)^0 = 1).
Fact 1 is INSUFFICIENT

Fact 2: W is even

This tells us nothing about the value of X.
Fact 2 is INSUFFICIENT

Combined, we know....
X^W = 1
W is EVEN

IF....
W = 0, then X = ANY positive integer (1^0 = 2^0 = 3^0, = 1 etc.)
W = 2, 4, 6, 8, etc. then X = 1 (1^2 = 1^4 = 1^6 = 1^8 = 1 etc.)
W = -2, then X = 1 since (1^(-2) = 1)
So, W can be ANY EVEN INTEGER and X can be ANY positive integer.
Combined, INSUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

e
x^w=1: x can be any value with w as 0 and x 1 can take w any value not sufficient
w even nothng from it
combined w =0 can give different values for x so not sufficient

VERITAS PREP OFFICIAL SOLUTION

Correct Answer: E

Any number raised to the zero power results in 1, so statement (1) tells us that w = 0 (in which case x could be anything) or that x = 1 (in which case w can be anything); not sufficient. Statement (2) tells us nothing about x; not sufficient. Combining the two statements, we learn that w still could be 0, since zero is an even integer, or x could be 1. But in the case that w = 0, x could still be any number, so both statements together are not sufficient.

from stmt 2 we cant say anything hence not sufficient. From satemet 1 a could be any number ...combining also we get w as event number may be 0 then staemnt 1 x can be any number from 1 to infinity hence E.

Target question: What is the value of positive integer x?

Statement 1: x^w=1
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x and w that satisfy statement 1. Here are two:
Case a: x = 1 and w = 2. In this case, the answer to the target question is x = 1
Case b: x = 2 and w = 0. In this case, the answer to the target question is x = 2
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of testing values when a statement doesn't feel sufficient, read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: w is even
Since we have no information about x, it's impossible to answer the target question with certainty
So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
At this point, I recognize that we can re-use the values we used to show that statement one is not sufficient, since w is even in both of those cases.
So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: x = 1 and w = 2. In this case, the answer to the target question is x = 1
Case b: x = 2 and w = 0. In this case, the answer to the target question is x = 2
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

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