Hi j_shreyans,

In Roman Numeral questions, it's often easiest to DISPROVE the Roman Numerals, if possible (so that you can eliminate answer choices).

Here, we're told that the MEAN of 4 integers will NOT change if each integer is multiplied by the same (ANY) constant. We're asked which of the Roman Numerals is ALWAYS TRUE.

RN II: The sum of the largest and smallest is 0

IF...the set is {-3, 0, 1, 2} then the mean would stay the same, but the sum of the largest and smallest is NOT 0.
RN II is NOT true.
Eliminate Answers B, D and E

RN III: The set contains BOTH positive AND negative numbers

IF....the set is {0, 0, 0, 0} then the mean would stay the same and the set does NOT contain both positive and negative numbers.
RN III is NOT true.
Eliminate Answer C.

There's only one answer left....

Final Answer: A

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

I'd like to add something to what Rich summed up nicely....

It's quite possible that your confusion about this question came from the ambiguous wording. "Multiplied by any constant" should really read "multiplied by the SAME constant" to avoid ambiguity. If each term could be multiplied by ANY constant, but they could be DIFFERENT, then the only possibility would be that each of the 4 terms is equal to 0.

A real GMAT question would have more precise language!