I am posting the question in the proper formatting.

If the square root of P² is an integer greater than 1, which of the following must be true?

I. P² has an odd number of positive factors

II. P² can be expressed as the product of an even number of positive prime factors

III. P² has an even number of positive factors

richachampion wrote:I am posting the question in the proper formatting.

If the square root of P² is an integer greater than 1, which of the following must be true?

I. P² has an odd number of positive factors

square root of P² is an integer greater than 1 - This means that P is a perfect square and all perfect square has odd number of +ve factors. CORRECT!

II. P² can be expressed as the product of an even number of positive prime factors

Any Perfect square can be expressed as the product of an even number of positive prime factors. CORRECT!

III. P² has an even number of positive factors

Since P is a perfect square that means this statement is not true as perfect square has odd number of factors.

The above gives us D, but what about P=0? I believe that this is a Poor question.

Hi Needgmat,

Below three points will be helpful for you, if you are solving questions about the perfect square.
 1. A perfect square will always have an odd number of factors
 2. A square of a prime number will always have exactly 3 factors
 3. A perfect square will have an odd number of odd factors and even number of even factors

From the first line of the question, it's clear that it is a perfect square.

Also Basic rule for a perfect square is that,

If N is a perfect square and expressed as p^x * q^y * r^z ....(where p, q and r are prime numbers) Then x , y and z are divisible by 2.

Also this is a must be question.

Remember to Plug in.

I. P² has an odd number of positive factors
- this goes by the rule no.1(A perfect square will always have an odd number of factors) or just plug in 36 is a perfect square where the number of factors are odd = 9. So true. So eliminate answer choices B, C and E.

II. P² can be expressed as the product of an even number of positive prime factors .
This goes by the definition of the perfect square. So true. Eliminate the answer choice A. Answer has to be D. no need to check the third statement. Plug in No need here.

III. P² has an even number of positive factors - this is against the first rule((A perfect square will always have an odd number of factors)) mentioned above. So this is not true.

So the answer is D.

Hope this helps.

Would like to know the source of this question? Its poorly framed because statement I and III are just opposite to each other.

We might want to go a little deeper into some of the logic here.

I:

Take a perfect square such as 36. Now think of it as the product of a pair of its factors:

1 * 36
2 * 18
3 * 12
4 * 9
6 * itself

Notice how we can pair the factors off in twos, except for the root, which can only be paired with itself? So we'll have 2*(# of pairs) + (the root), or 2*(some integer) + 1 unique factors. 2*integer + 1 is odd, so we'll have an odd number of factors.

II:

p² = p * p. Suppose that p has n prime factors. That means p * p has 2n prime factors, so p² = the product of an even number of prime factors.

(For instance, say p = 44100. Then p = (2*3*5*7) * (2*3*5*7), which has 2*4 prime factors, or 8 prime factors.)

III:

Since I is true, III must be false. This one seems to be testing whether you're still awake after sorting out I and II