Perfect square

[pavenleo]

If the square root of p2 is an integer, which of the following must be true?

I. p2 has an odd number of factors
II. p2 can be expressed as the product of an even number of prime factors
III. p has an even number of factors

Here is explanation from Manhattan GMAT CAT:
If the square root of p2 is an integer, p is a perfect square. Let’s take a look at 36, an example of a perfect square to extrapolate some general rules about the properties of perfect squares.

Statement I: 36’s factors can be listed by considering pairs of factors (1, 36) (2, 18) (3,12) (4, 9) (6, 6). We can see that they are 9 in number. In fact, for any perfect square, the number of factors will always be odd. This stems from the fact that factors can always be listed in pairs, as we have done above. For perfect squares, however, one of the pairs of factors will have an identical pair, such as the (6,6) for 36. The existence of this "identical pair" will always make the number of factors odd for any perfect square. Any number that is not a perfect square will automatically have an even number of factors. Statement I must be true.


My question is related to the bold phrase above. Why we dont count negative integer factors: 36 also can be written as -1*(-36); -2*(-18); -3*(-12).....So combine negative factors and positive factors we have an even number of factors. Please explain why you only count the postive one?

Thanks

[RonPurewal]

Technically you're right; it should say "positive factors".

GMAC has never tested "negative factors", so, if you ever see a problem that doesn't specify, you should go ahead and assume that we're talking about positive factors. But I'll submit the problem for revision.

[pavenleo]

Thank you very much Ron!

[RonPurewal]

Ok, we updated the problem.

Thank you very much Ron!

Sure. Thanks to you for pointing out the issue.

[drtfyghujd403]

Can anyone tell me what is the final answer? With explanations of b) and c)?

[tim]

I take it you're asking this because you saw the problem here but haven't encountered the question in one of your CAT exams (if you saw it there, read the explanation and let us know if you have any further questions). I'd like you to try this question yourself and let us know what you are able to come up with and where you get stuck, then we'll be glad to help you from there.

[PuravS276]

I've looked at the answer explanation in the CAT. It says nothing about if p=1. If we have p=1 then only statement 1 should be true. Please let me know if this is correct.

[RonPurewal]

you have a valid point.

and, along the same lines, there's also p = 0, which technically fits the condition (√p = integer) but does not have any sort of canonical prime factorization.

i've updated the question so that it now excludes these cases. (everything about the question remains the same, except it now says that √p is an integer greater than 1 (as opposed to simply 'an integer').

[VivekG483]

The answer choice for this question still seem confusing:

II. p2 can be expressed as the product of an even number of positive prime factors

Can't, 36 be also expressed as (-6)^2*1^2, hence an even number of odd prime factors!!.. Am i thinking it right? This will make option 2 not always correct!

[RonPurewal]

you're making at least three errors here.

• primes are NEVER negative.
when the problems say 'positive prime numbers' they are just being redundant—just to be extra sure that people don't start trying to get 'tricky'.

• 6 is not a prime number.

• 1 is not a prime number.