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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]

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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]

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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.

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Re: If root(x) is a positive integer is root(x) a prime number? [#permalink]

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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]

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18 Dec 2015, 16:45
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Re: If root(x) is a positive integer is root(x) a prime number? [#permalink]

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18 Dec 2015, 20:25
St 1 is enough as we know prime has 2 factor. if prime is squared it has 3 factors. i.e. consider 25 it has 1,5 and 25.

St 2 it says all factors, consider multiple of odd 3x5=15
lets say 225 is x. where it is not prime.
if we consider x as 25 it gives a prime no.

so clearly st 1 is enough.

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Re: If root(x) is a positive integer is root(x) a prime number?   [#permalink] 18 Dec 2015, 20:25
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