Hi GMATsid2016,
This Roman Numeral question is based on a series of Number Property rules, but you don't need to know the rules to get the correct answer - you can TEST VALUES and do some brute-force math.
From the prompt, we know that P is a positive INTEGER greater than 1. We're asked which of the 3 Roman Numerals MUST be true. In most Roman Numeral questions, the 'key' is to DISPROVE the Roman Numerals so that we can quickly eliminate answer choices. Here though, we're going to prove that patterns exist.
Since P is a positive integer, we know that P^2 is a perfect square.
I. P^2 has an odd number of positive factors
IF...
P = 2, P^2 = 4 and the factors are 1, 2 and 4... so there IS an odd number of factors
P = 3, P^2 = 9 and the factors are 1, 3 and 9... so there IS an odd number of factors
P = 4, P^2 = 16 and the factors are 1, 2, 4, 8 and 16... so there IS an odd number of factors
Notice the pattern here. Since P^2 is a perfect square, there will ALWAYS be an odd number of factors, so Roman Numeral 1 IS true.
Eliminate Answers B, C and E.
From the answers that remain, we only have to deal with Roman Numeral II.
II. P^2 can be expressed as the product of an even number of positive prime factors
Using the same examples from Roman Numeral I, you can prove this pattern too:
P = 2, P^2 = 4 and we can get to 4 by multiplying (2)(2)... an even number of positive prime factors
P = 3, P^2 = 9 and we can get to 9 by multiplying (3)(3)... an even number of positive prime factors
P = 4, P^2 = 16 and we can get to 16 by multiplying (2)(2(2)(2))... an even number of positive prime factors
Since P^2 is a perfect square there will ALWAYS be a product of positive prime factors that will end in P^2, so Roman Numeral 2 IS true.
Eliminate Answer A.
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
