Hi j_shreyans,
This question is perfect for TESTing VALUES, but it's going to take a lot more work than usual to get to the correct answer. Roman Numeral questions are relatively rare on Test Day (you'll probably see just 1-2 in the Quant; in rare cases, they can show up in an RC question too). While it's common for these questions to take a bit more time/work than average, it's also quite common for you to be able to use the answer choices and the frequencies of the Roman Numerals to avoid some of the work. I'm going to work through all three Roman Numerals though...
We're told that N is a positive integer and a PERFECT SQUARE, which is a rather limiting piece of information.
A quick list of the first several positive perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100....
We're asked which of the following MUST be true, which really means "which of the following is ALWAYS TRUE no matter how many different examples you try?"
Sometimes the easiest way to get to the answer is to do some brute-force math....
I. The NUMBER of DISTINCT factors of N is ODD.
Let's count up factors....
If....
N = 1, there is just 1 factor. 1 = ODD
N = 4, factors are 1, 2 and 4. 3 total = ODD
N = 9, factors are 1, 3 and 9. 3 total = ODD
N = 16, factors are 1, 2, 4, 8 and 16. 5 total = ODD
N = 25, factors are 1, 5 and 25. 3 total = ODD
N = 36, factors are 1, 2, 3, 4, 6, 9, 12, 18, 36. 9 total = ODD
N = 49, factors are 1, 7, 49. 3 total = ODD
At this point, I don't think that we're dealing with a co-incidence. The GMAT doesn't expect us to keep working forever, nor is it likely that some far-off number is an exception to this pattern (the GMAT writers don't "work" that way).
#1 appears to be TRUE.
II. The SUM of the DISTINCT factors of N is ODD.
Thankfully, the list that we put together for Roman Numeral 1 is right in front of us, so now we can do some simple addition...
If...
N = 1, the sum = 1 = ODD.
N = 4, 1+2+4 = 7 = ODD.
N = 9, 1+3+9 = 13 = ODD.
N = 16, 1+2+4+8+16 = 31 = ODD.
N = 25, 1+5+25 = 31 = ODD.
N = 36, 1+2+3+4+6+9+12+18+36 = 91 = ODD
N = 49, 1 + 7 + 49 = 57 = ODD
Just as we saw in Roman Numeral 1, this appears to be a pattern.
#2 appears to be TRUE.
III. The NUMBER of DISTINCT PRIME factors of N is EVEN.
If....
N = 1, there are 0 primes = EVEN
N = 4, there is 1 prime (the number 2) = ODD
Roman Numeral III is NOT always true.
Final Answer: C - I and II
GMAT assassins aren't born, they're made,
Rich
