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If root(x) is a positive integer is root(x) a prime number?

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If root(x) is a positive integer is root(x) a prime number? [#permalink] New post 05 Dec 2012, 06:24
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If \sqrt{x} is a positive integer, is \sqrt{x} a prime number?

(1) x is divisible by exactly 3 positive integers
(2) All positive factors of x are odd
[Reveal] Spoiler: OA

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Last edited by Bunuel on 05 Dec 2012, 06:30, edited 1 time in total.
Renamed the topic and edited the question.
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Re: If root(x) is a positive integer is root(x) a prime number? [#permalink] New post 05 Dec 2012, 06:42
If \sqrt{x} is a positive integer, is \sqrt{x} a prime number?

(1) x is divisible by exactly 3 positive integers. The fact that x has exactly 3 factors means that x=prime^2 (in this case the number of factors will be 2+1=3: 1, prime, and x. Check here: math-number-theory-88376.html). Therefore, \sqrt{x}=\sqrt{prime^2}=prime. Sufficient.

(2) All positive factors of x are odd. If x=1, then \sqrt{x}=1\neq{prime} but if x=9 (9 has 3 odd factors: 1, 3, and 9), then \sqrt{x}=3={prime}. Not sufficient.

Answer: A.
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Re: If root(x) is a positive integer is root(x) a prime number? [#permalink] New post 05 Dec 2012, 06:57
Thanks for the clarification..
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Re: If root(x) is a positive integer is root(x) a prime number?   [#permalink] 05 Dec 2012, 06:57
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