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If a and b are positive integers, is ³√ab an an integer?

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If a and b are positive integers, is ³√ab an an integer?  [#permalink]

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New post Updated on: 24 Oct 2017, 21:18
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If a and b are positive integers, is \(\sqrt[3]{ab}\) an an integer?

(1) \(\sqrt{a}\) is an integer

(2) \(b=\sqrt{a}\)


I think this is from OG 2017 quant review, I found it in "New questions from OG 17 Review" file that was shared here by a member. So, I don't have the OA. If someone could please post the OA and solution, would be appreciated.

Thanks.

Originally posted by Neeraj91 on 15 Aug 2016, 07:53.
Last edited by Bunuel on 24 Oct 2017, 21:18, edited 2 times in total.
Edited the question and added the OA.
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Re: If a and b are positive integers, is ³√ab an an integer?  [#permalink]

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New post 15 Aug 2016, 08:10
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If a and b are positive integers, is \(\sqrt[3]{ab}\) an an integer?

(1) \(\sqrt{a}\) is an integer. No info about b. Not sufficient.

(2) \(b=\sqrt{a}\) --> square: \(b^2 = a\) --> substitute: \(\sqrt[3]{ab}=\sqrt[3]{b^2*b}=b=integer\). Sufficient.

Answer: B.
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Re: If a and b are positive integers, is ³√ab an an integer?  [#permalink]

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New post 15 Aug 2016, 08:57
Aah. I chose E, since i substituted the value for b in the original question, to get a^1/7. Didn't think of just squaring it! Thanks Bunnuel!
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Re: If a and b are positive integers, is ³√ab an an integer?  [#permalink]

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New post 20 Aug 2016, 05:39
1
Neeraj91 wrote:
If a and b are positive integers, is \(\sqrt[3]{ab}\) an an integer?

(1) \(\sqrt{a}\) is an integer

(2) \(b=\sqrt{a}\)


I think this is from OG 2017 quant review, I found it in "New questions from OG 17 Review" file that was shared here by a member. So, I don't have the OA. If someone could please post the OA and solution, would be appreciated.

Thanks.


1. If a=4 and b=2 then Yes If a=4 and b=1 then No. Insufficient
2. Since both a and b are positive we can square both sides and get a=(b^2) Substitute to have \(\sqrt[3]{b^3}\) Sufficient
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Re: If a and b are positive integers, is ³√ab an an integer?  [#permalink]

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New post 30 May 2017, 01:06
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I have a question.
No doubt that stat1 is not sufficient.
BUT - if I take stat2 and replace b by SQRT(a) in the original question, I get (a*SQRT(a))^(1/3) = (a^(3/2))^(1/3) = SQRT(a).
In this case, I will need both statements to have a definitive answer and thus, the answer is C.
How can I differentiate between my reasoning and Bunuel's?
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Re: If a and b are positive integers, is ³√ab an an integer?  [#permalink]

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New post 30 May 2017, 01:16
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Mike2805 wrote:
I have a question.
No doubt that stat1 is not sufficient.
BUT - if I take stat2 and replace b by SQRT(a) in the original question, I get (a*SQRT(a))^(1/3) = (a^(3/2))^(1/3) = SQRT(a).
In this case, I will need both statements to have a definitive answer and thus, the answer is C.
How can I differentiate between my reasoning and Bunuel's?


We are given that a and b are positive integers. (2) says that \(b=\sqrt{a}\), thus \(b=\sqrt{a}=integer\). So, when you get \(\sqrt{a}\), you already know that it must be an integer.

Hope it's clear.
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If a and b are positive integers, is ³√ab an an integer?  [#permalink]

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New post 13 Nov 2018, 18:39
This question tricks your mind, your brains seems to follow thought process applied in statement I, and see statement II in the same light and you go for the easier option "C" and do not exert much force on proving B to be sufficient.

Seems like, one should start with statement II, in higher level questions.
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Re: If a and b are positive integers, is ³√ab an an integer?  [#permalink]

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New post 14 Nov 2018, 09:13
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If a and b are positive integers, is ∛ab an integer?

(1) √a is an integer

(2) b= √a

The question is invariably asking us to determine whether the expression inside the radical, ab, is a perfect cube. One basic property of a perfect cube which will be very useful in problems like these, is that every perfect cube contains a triplet of prime factors. For example,

8 = 2* 2*2; 125 = 5*5*5; 216 = 2*2*2*3*3*3

Hence, in this question,we should be able to find out if the expression ab can give us triplet/s of prime factors.

The first statement, when considered alone, is quite obviously insufficient to answer the question with a definite YES or a definite NO since it does not provide any information about 'b'. The only conclusion we can draw from this statement is that 'a' is a perfect square. So, a can take values of 1 or 4 or 9 and so on. However, 'ab' being a perfect cube depends on the specific value of 'b' which is why the first statement is not sufficient when taken alone. For example,

If a=4 and b=2, ab = 8 is a perfect cube and we can answer the main question with a Yes. However, if a=4 and b=3, ab= 12 which is not a perfect cube and we can answer the main question with a No.

From the second statement, we can gather that 'a' should be a perfect square since the square root of 'a' is giving us 'b', which is also an integer (given in the question statement). Also, we may say that b^2=a since b= √a (squaring both sides). Therefore, ab = b^3 which is definitely a perfect cube since 'b' is an integer. As such, the second statement alone is sufficient to answer the question asked with a definite Yes.

The easiest trap that one can fall into in such questions, is to assume that the two statements have to be combined to find out the answer. This can happen when you have not fully processed the data given in the question statement and hence not retained it during the solution.

Option B is the answer.

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Re: If a and b are positive integers, is ³√ab an an integer? &nbs [#permalink] 14 Nov 2018, 09:13
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