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If x is a positive integer, is √√x an integer? (1) x is the square of

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danzig
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If x is a positive integer, is √√x an integer? (1) x is the square of [#permalink] New post   Updated on: 21 May 2021, 05:12
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If x is a positive integer, is √√x an integer?

(1) x is the square of an integer.

(2) √x is the square of an integer.

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I agree with the OA. However, I have a doubt with (2):

\(\sqrt{x} = i^2\) , being \(i\) an integer.

Should we unsquare or use an additional square root on \(i\)? Is there a difference? I know that the square root only provide the positive root of the number.

Thanks!
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Originally posted by danzig on 29 Jul 2013, 16:18.
Last edited by Bunuel on 21 May 2021, 05:12, edited 3 times in total.
Edited the question.
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If x is a positive integer, is √√x an integer? (1) x is the square of [#permalink] New post   03 Aug 2013, 08:46
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The question asks if \(x^{\frac{1}{4}}\) is a positive integer

(1) x is the square of an integer.
So x can be \(4\), and \(4^{\frac{1}{4}}\) is NOT a positive integer.
But x can be \(16\), and \(16^{\frac{1}{4}}\) is a positive integer.
Not sufficient

(2) \(\sqrt{x}\) is the square of an integer.
So \(\sqrt{x}=n^2\), \(x^{\frac{1}{4}}=n^{2*\frac{1}{2}}=n\) where n is an integer. Sufficient

Hope it's clear
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Re: If x is a positive integer, is √√x an integer? (1) x is the square of [#permalink] New post   30 Mar 2019, 11:23
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Using Statement 1, if x is the square of some integer n, then x = n^2. So we want to know if √√(n^2) = √n is an integer. We don't know, since we don't know if n itself is the square of an integer.

Using Statement 2, if √x is the square of some integer m, then √x = m^2. So we want to know if √√x = √(m^2) = m is an integer, and it is. So Statement 2 is sufficient.
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Re: If x is a positive integer, is √√x an integer? (1) x is the square of [#permalink] New post   30 Mar 2019, 12:07
Statement 1: INSUFFICIENT
Replacing x by square of an integer, the question becomes: ✓✓(square of a positive integer)
Or, ✓(a positive integer) = an integer?

Answer may be yes or NO.

IN CASE OF NUMBER PICKING:
if x=16, ✓✓16=2= integer (yes to main question)

if x=4, ✓✓4=✓2= NOT integer (NO to main question)

STATEMENT 2: SUFFICIENT

Replacing ✓x by square of an integer, the question becomes: ✓(square of a positive integer)
= (a positive integer) = an integer (YES TO MAIN QUESTION)

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Re: If x is a positive integer, is √√x an integer? (1) x is the square of [#permalink]
30 Mar 2019, 12:07
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Bunuel
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