Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors.

Tips about the perfect square:

1. The number of distinct factors of a perfect square is ALWAYS ODD.
2. The sum of distinct factors of a perfect square is ALWAYS ODD.
3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
4. Perfect square always has even powers of its prime factors.

So I and II must be true.
_________________

Thanks for the tips Bunuel. I had never thought about 2 and 3, but they make sense.

To clarify, this requires the prime factorization of a number.

Thank you! Bunuel & g4gmat for sharing your tips.

Those are useful properties which are worth to remember, even better if you understand why they are right.

Questions about these properties with explanation why they are right:
help-factors-problem-99145.html?hilit=perfect%20square
perfect-square-94700.html?hilit=perfect%20square

As for picking numbers: you can easily prove that III is not always true as soon as you pick appropriate perfect square, say n=2^2=4 --> 4 has 1 (so odd) prime factor, which is 2. For I and II if you try 2-3 perfect squares you'll see that all of them will have the odd number of distinct factors and the odd sum of the distinct factors and though 2-3 examples do not prove that these statement are ALWAYS true you can make educated guess.

The question asks which of the following MUST be true, or which of the following is ALWAYS true no matter what set of numbers you choose. Generally for such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

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.

Hope it helps.
_________________

These two facts about perfect square is applicable in this problem, so the Option I and II is true

1. The number of distinct factors of a perfect square is ALWAYS ODD.
2. The sum of distinct factors of a perfect square is ALWAYS ODD

With reference to point#2, though "The sum of distinct factors of a perfect square is ALWAYS ODD", the vice versa may not be true. Consider the number 2 (factors 1 & 2) and number 8 (factors 1, 2, 4, & 8) -- these are not perfect squares but sum of their distinct factors are odd.

Is 0 considered a perfect square?

That is what I wanted to point out - that those properties do not apply to 0. Thank you for confirming. However, the question being discussed mentions 'a positive integer', so it should be fine.

Factors of 49 are 1, 7, and 49: 1 + 7 + 49 = 57 = odd.
_________________