IMO answer should be (D).

Can you please recheck your OA. Explanation after confirmation.

Notice that x^2 - y^2 = (x + y)(x - y). This is a commonly known formula, so I advise you to remember it.

1. x + y is odd helps partially, but it's not enough. You get that x^2 - y^2 = (odd number)*(unknown number). Since x - y can be either odd or even (and depending on that x^2 - y^2 will be odd or even), 1 is insufficient

2. again, you only have part of the picture: x^2 - y^2 = (unknown number)*(odd number). Using the same line of thought as above, 2 is insufficient.

But put both stmts together and get that x^2 - y^2 = (odd number)*(odd number), which will always be odd. This is why the answer is C[/spoiler]

Agree with Code on D) inspite of the OA supposedly being C

X^2-Y^2 = (X+Y) (X-Y)

Stmt I

x+y is odd

odd+even = odd
even+odd = odd

If x+y is odd x-y must also be odd therefore

X^2-Y^2 = (X+Y) (X-Y)
= odd*odd
= odd

Stmt II

Similar explanation to I

Choose D

Must be missing something if the OA is correct and turns out be C

DanaJ wrote:Notice that x^2 - y^2 = (x + y)(x - y). This is a commonly known formula, so I advise you to remember it.

Okay lets look at this way....

ODD-EVEN Rules

even + even = even
even + odd = odd
odd + even = odd
odd + odd = even

even - even = even
even - odd = odd
odd - even = odd
odd - odd = even

odd^2 = odd
even^2 = even

odd^2 - odd^2 = odd - odd = even
odd^2 - even^2 = odd - even = odd
even^2 - odd^2 = even - odd = odd
even^2 - even^2 = even - even = even

Now let's solve real question,

As per statement 1,

X + Y is ODD

that means following two rules are applicable:-
even + odd = odd OR
odd + even = odd

Now it means if X is ODD then Y must be EVEN or vice-versa.

So if X is ODD then Y must be EVEN.
i.e. ODD^2 - EVEN^2 = ODD - EVEN = ODD

or if Y is ODD then X must be EVEN.
i.e. EVEN^2 - ODD^2 = EVEN - ODD = ODD

So both cases results in ODD answer. Hence statement I should be alone sufficient.

Now let's take statement II,

X - Y = ODD

that means following two rules are applicable:-
even - odd = odd OR
odd - even = odd

Now it means if X is ODD then Y must be EVEN or vice-versa.

So if X is ODD then Y must be EVEN.
i.e. ODD^2 - EVEN^2 = ODD - EVEN = ODD

or if Y is ODD then X must be EVEN.
i.e. EVEN^2 - ODD^2 = EVEN - ODD = ODD

So both cases results in ODD answer. Hence statement II also should be alone sufficient.

Since both statements alone are suffice to deduce the result hence IMO OA should be (D) rather than (C).

Any thoughts....

The only technicality I did miss was x and y doesnt have to be integers

Stmt I

x=1.5 y=1.5

x+y-> odd and x^2-y^2 is even

x=7 y=2

x+y odd and x^2-y^2 is odd

INSUFF

Stmt II

x=4.5 y=1.5

x=7 y=2

INSUFF

Together:

odd*odd =odd

SUFF

Hence C

Nice one....

Regards,
CR

@cramya:
I didn't even think of that, but I guess that sometimes overthinking it hurts...

cramya wrote:The only technicality I did miss was x and y doesnt have to be integers

Hmm.... thanks CR.