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# Is n the square of an integer? (1) n is the square root of an integer

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Math Expert
Joined: 02 Sep 2009
Posts: 51280
Is n the square of an integer? (1) n is the square root of an integer  [#permalink]

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01 Jul 2018, 20:47
1
9
00:00

Difficulty:

95% (hard)

Question Stats:

20% (02:03) correct 80% (01:37) wrong based on 116 sessions

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Is n the square of an integer?

(1) n is the square root of an integer
(2) $$\sqrt{9n}$$ is square of an integer

_________________
examPAL Representative
Joined: 07 Dec 2017
Posts: 842
Is n the square of an integer? (1) n is the square root of an integer  [#permalink]

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Updated on: 01 Jul 2018, 23:13
1
1
Bunuel wrote:
Is n the square of an integer?

(1) n is the square root of an integer
(2) $$\sqrt{9n}$$ is square of an integer

Questions dealing with number properties can often be solved with very little calculations, using logic only.
We'll look for such a solution, a logical approach.

(1) Every positive integer is the square root of another integer (i.e x is the sqrt of x^2). But this doesn't mean that it also has to be the square of an integer!
Insufficient

(2) So $$\sqrt{9n}=3\sqrt{n}$$ is the square of an integer meaning that $$\sqrt{n}$$ can be written as k/3 for some integer k. That means that n = k^2/9 for some integer k. This is an integer only if k is divisible by 3, which it does not have to be.
Insufficient.

Combined:
So from (1) we know that n^2 must be an integer, and from (2) we know that n = k^2/9 for some integer k.
Combining, if n=k^2/9 then n^2 = k^4/81. Since this must be an integer then k^4 is divisible by 81 so k^2 is divisible by 9 and k is divisible by 3. Therefore n must be an integer.

(C) is our answer.
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Originally posted by DavidTutorexamPAL on 01 Jul 2018, 21:38.
Last edited by DavidTutorexamPAL on 01 Jul 2018, 23:13, edited 1 time in total.
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Joined: 04 Apr 2015
Posts: 107
Re: Is n the square of an integer? (1) n is the square root of an integer  [#permalink]

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01 Jul 2018, 21:58
1
DavidTutorexamPAL wrote:
Bunuel wrote:
Is n the square of an integer?

(1) n is the square root of an integer
(2) $$\sqrt{9n}$$ is square of an integer

Questions dealing with number properties can often be solved with very little calculations, using logic only.
We'll look for such a solution, a logical approach.

(1) Every positive integer is the square root of another integer (i.e x is the sqrt of x^2). But this doesn't mean that it also has to be the square of an integer!
Insufficient

(2) So $$\sqrt{9n}=3\sqrt{n}$$ is the square of an integer meaning it is also an integer. Then $$\sqrt{n}$$ must be an integer meaning that n must be a square of an integer.
Sufficient.

(B) is our answer.

Answer should be C

if n =4/9 statement 2 still holds.

Waiting for the OA
examPAL Representative
Joined: 07 Dec 2017
Posts: 842
Re: Is n the square of an integer? (1) n is the square root of an integer  [#permalink]

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01 Jul 2018, 23:13
varundixitmro2512 wrote:
DavidTutorexamPAL wrote:
Bunuel wrote:
Is n the square of an integer?

(1) n is the square root of an integer
(2) $$\sqrt{9n}$$ is square of an integer

Questions dealing with number properties can often be solved with very little calculations, using logic only.
We'll look for such a solution, a logical approach.

(1) Every positive integer is the square root of another integer (i.e x is the sqrt of x^2). But this doesn't mean that it also has to be the square of an integer!
Insufficient

(2) So $$\sqrt{9n}=3\sqrt{n}$$ is the square of an integer meaning it is also an integer. Then $$\sqrt{n}$$ must be an integer meaning that n must be a square of an integer.
Sufficient.

(B) is our answer.

Answer should be C

if n =4/9 statement 2 still holds.

Waiting for the OA

You're right! My bad, fixed.
_________________

Claim your offer here.

Happy holidays!

Re: Is n the square of an integer? (1) n is the square root of an integer &nbs [#permalink] 01 Jul 2018, 23:13
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# Is n the square of an integer? (1) n is the square root of an integer

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