If a and b are distinct non zero numbers, is x+y a

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If a and b are distinct non zero numbers, is x+y an even integer ?

(1) a^x*a^y = 1.
(2) b^x*b^y = 1.

How to determine the statements suffieciency? Can some experts help?

OA C

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Hi leihannie07,

We're told that A and B are DISTINCT non-0 numbers. We're asked if X+Y is an EVEN integer. This is a YES/NO question that can be solved by TESTing VALUES.

1) (A^X)(A^Y) = 1

To start, it's important to note that we can 'rewrite' this equation as (A)^(X+Y) = 1

IF...
A=1, then X and Y can be ANY VALUES. Thus, it's possible that X+Y could be an even integer (a 'YES' answer), an odd integer (a 'NO' answer) or a non-integer (also a 'NO' answer).
Fact 1 is INSUFFICIENT

2) (B^X)(B^Y) = 1

The same logic that applies to Fact 1 also applies to Fact 2. We can rewrite this equation as (B)^(X+Y) = 1

IF...
B=1, then X and Y can be ANY VALUES. Thus, it's possible that X+Y could be an even integer (a 'YES' answer), an odd integer (a 'NO' answer) or a non-integer (also a 'NO' answer).
Fact 2 is INSUFFICIENT

Combined, we know that that A and B must be DISTINCT (meaning 'different') so while one of those variables could equal 1, the other would have to equal something else. Thus, we have to focus on that 'non 1' value and see what could happen...

IF...
A = -1, then X and Y would have to either be BOTH EVEN or BOTH ODD. (Even + Even = Even, so the answer is 'YES') and (Odd + Odd = Even, so the answer is also 'YES').
A = anything integer other than -1, 0, or 1, then (X+Y) would have to equal 0 (since any non-0 value raised to the '0 power' equals 1) and the answer would also be 'YES').
The answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

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