If the square root of p2 is an integer, which

[jasonk1234]

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

I
II
III
I and II
II and III


I had used "0" thinking that the square root of 0^2 would be 0. I thought this would disprove Statement II. Where am I going wrong? Am I not able to use 0 Because the Square root of zero is undefined? Or is that their are 0 prime factors and 0 is even? Please let me know the flaw in my logic. I appreciate everyones help.

[RonPurewal]

Huh.

If "p2" means p squared (which we would normally write on here as p^2), then the problem statement is telling you only that the absolute value of p is an integer (because root (p^2) is the absolute value of p), which means, in short, that p is an integer.

So the problem is just, "If p is an integer, then which of the following is true?"

I agree with everything you've said about choice II. Yes, you are correct: the square root of 0 is 0, and 0 can't be reduced to a product of primes.
Incidentally, p = 1 and p = -1 are also problematic, for the same reason (p^2 would be 1, which also lacks a prime factorization).

The gist of the problem is that, for any OTHER p, p^2 DOES factor into an even # of primes (it's a perfect square, so you get two copies of everything in the factorization). I'll submit this problem for 'fixage'.

Other answers:
I isn't true. If p is 3, for instance, then p^2 is 9, which has three factors (1, 3, and 9).
III isn't true either, since p can be any integer at all. (Therefore, the example of p=3 is a counterexample again.)

[srikant]

Ron,

Did you get the question changed?
Please let me know how did you rephrase it or what changes did you make?

I had selected option 2 as I never thought of using 0 for P.

I think if we can fix this problem by using positive integer instead of just integer and then option 2 would be the correct one.

[drtfyghujd403]

Can anyone tell me what the final answer is with explanation please?

[tim]

You asked this same question in another thread. Please read my answer there.