Question

A question is followed by two statements, numbered (1) and (2). Using the information provided and general knowledge, decide whether the information given is sufficient to solve the problem.
Is \(x^2-y^2\) even?
(1) \(x+y\) is even.
(2) \(x-y\) is odd.

• A. Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer 1

The Correct Option is A

From statement 1: 
x + y = Even 
\(x^2 – y^2 = (x + y)(x – y) = (Even) \times (x – y)\)
When an even number is multiplied by another even number, the result is always an even and when an even number is multiplied by an odd number, again the result is always an even. Therefore, we get a unique answer from statement 1 alone. 
So, statement 1 alone is sufficient.
From statement 2:
x – y = Odd 
\(x^2 – y^2 = (x + y)(x – y) = (x + y) (Odd) \)
When an odd number is multiplied by another odd number, the result is always an odd number while when an odd number is multiplied by an even number, the result is always an even number. Therefore, we do not get a unique answer from statement 2. 
So, statement 2 alone is not sufficient.
So, the correct answer is A.