If the square root of \(p^2\) is an integer greater than 1, which of the following must be true?
I. \(p^2\) has an odd number of positive factors
II. \(p^2\) can be expressed as the product of an even number of positive prime factors
III. \(p\) has an even number of positive factors
A. I
B. II
C. III
D. I and II
E. II and III
[spoiler]OA=D[/spoiler]
Source: Manhattan GMAT
If the square root of \(p^2\) is an integer greater than 1, which of the following must be true?
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Re: If the square root of \(p^2\) is an integer greater than 1, which of the following must be true?
Since p^2 > 1, p > 1 or p < -1. Let’s consider p > 1 first.Vincen wrote: ↑Mon May 18, 2020 7:03 amIf the square root of \(p^2\) is an integer greater than 1, which of the following must be true?
I. \(p^2\) has an odd number of positive factors
II. \(p^2\) can be expressed as the product of an even number of positive prime factors
III. \(p\) has an even number of positive factors
A. I
B. II
C. III
D. I and II
E. II and III
[spoiler]OA=D[/spoiler]
If p > 1, then p^2 always has an odd number of positive factors. So statement I is true. Statement II is also true since every distinct prime factor of p^2 must come in pairs or an even amount. However, statement III is not necessarily true since p itself can be a perfect square so p could have an odd number of positive factors.
Now, let’s consider p < -1.
If p < -1, statements I and II will still be true since p^2 = |p|^2 (notice that |p| > 1, so it behaves the same as p > 1). We don’t have to consider statement III because it’s proven false when p > 1 already.
Answer: D
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