If x + y is odd, which of the following could be true?

"MUST BE TRUE" questions:
These questions ask which of the following MUST be true, or which of the following is ALWAYS true for ALL valid sets of numbers you choose. Generally for such kind of questions if you can prove that a statement is NOT true for one particular valid set of numbers, it will mean that this statement is not always true and hence not a correct answer.

So, for "MUST BE TRUE" questions plug-in method is good to discard an option but not 100% sure thing to prove that an option is ALWAYS true.


As for "COULD BE TRUE" questions:
The questions asking which of the following COULD be true are different: if you can prove that a statement is true for one particular set of numbers, it will mean that this statement could be true and hence is a correct answer.

So, for "COULD BE TRUE" questions plug-in method is fine to prove that an option could be true. But here, if for some set of numbers you'll see that an option is not true, it won't mean that there does not exist some other set which will make this option true.

If x + y is odd, which of the following could be true?

(I) xy is even
(II) xy is odd
(III) x/y is even

A. I only
B. II only
C. III only
D. I and II
E. I and III

given X+Y is odd ( it means one is odd and other is even)
X Y XY
E O E
O E E
I True

2) XY - Odd, Not possible

3) X/Y is even
Let X be 2 and Y be 1 then X/Y = 2 even, it could be possible

Ans E

Asked: If x + y is odd, which of the following could be true?

x is even ; y is odd
or x is odd ; y is even

(I) xy is even: COULD BE TRUE
(II) xy is odd : NOT POSSIBLE
(III) x/y is even : COULD BE TRUE

IMO E

x + y = odd when x = even and y = odd or x = odd and y = even.

Even * ODD or ODD * EVEN = EVEN - Option I is TRUE and Option II is not TRUE.

\(\frac{Even }{ ODD}\) is EVEN but \(\frac{ODD }{ EVEN}\) = DECIMAL- Option III is TRUE and not TRUE.

Answer E

I have 1 doubt,

What if we take X=2 and Y=1/2;

• X+Y = 2.5 (Odd); Satisfies the given criteria
• Now, XY = 2*1/2 = 1 -> Which is Odd so II can be true

Is there any key element that I am missing?

Decimals cannot be classified as either odd or even, so you cannot consider 2.5 as odd. Only integers can be classified as odd or even.