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If a and b are positive integer,is (ab)^(1/3) an integer?

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If a and b are positive integer,is (ab)^(1/3) an integer?  [#permalink]

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If a and b are positive integer, is \(\sqrt[3]{ab}\) an integer?

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

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

OG Q 2017(Book Question: 300)

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Re: If a and b are positive integer,is (ab)^(1/3) an integer?  [#permalink]

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New post 24 Jun 2016, 12:32
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Statement 1: It is clearly insufficient as there is no information about b.

Statement 2: It is given in the question that a and b are integers. So, b=sqrt(a) is an integer.
(a*b)^1/3 = (a ^ (1+1/2))^(1/3) = (a^3/2)^(1/3) = a^(1/2) = sqrt (a) = b (b is an integer)
So, Statement 2 is sufficient.

Answer B!

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Re: If a and b are positive integer,is (ab)^(1/3) an integer?  [#permalink]

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New post 14 Oct 2016, 07:23
1
In this case:
3sqrt(ab)= integer, we need to know if the product of a and b is a cube of either a or b or any other number.

Case1: nothing is mentioned about one variable NS
Case2: B=sqrt(a)
hence B^2=A

Putting it in the given equation we have b^2*B= B^3.
Hence the cube root is an interger.
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Re: If a and b are positive integer,is (ab)^(1/3) an integer?  [#permalink]

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New post 15 Oct 2016, 06:54
AbdurRakib wrote:
If a and b are positive integer,is \(\sqrt[3]{ab}\) an integer?

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

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

OG Q 2017(Book Question: 300)


FROM STATEMENT - I

We will be unable to comment on \(\sqrt[3]{ab}\) from one variable \(\sqrt{a}\), we need information on the other variable \(b\) to ascertain the value.

FROM STATEMENT - II

This statement is sufficient , because we have the relationship between the two variable...

Hence , correct answer will be Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.


PS : Feel free to revert in case of any doubt..
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Re: If a and b are positive integer,is (ab)^(1/3) an integer?  [#permalink]

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New post 17 Jan 2017, 21:16
Abhishek009 wrote:
AbdurRakib wrote:
If a and b are positive integer,is \(\sqrt[3]{ab}\) an integer?

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

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

OG Q 2017(Book Question: 300)


FROM STATEMENT - I

We will be unable to comment on \(\sqrt[3]{ab}\) from one variable \(\sqrt{a}\), we need information on the other variable \(b\) to ascertain the value.

FROM STATEMENT - II

This statement is sufficient , because we have the relationship between the two variable...

Hence , correct answer will be Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.


PS : Feel free to revert in case of any doubt..


Thanks ..do you mean the specific relationship or just any relationship in your analysis of statement 2?
If we have the specific relationship which is described above, then shoud we not check by subsitituting in the prompt expression?

If for example I take another relationship ..say a= b/2..
cubroot(ab) = cubroot(b^2/2)
now let us take b = 4 .. b^2 = 16 and cubroot(b^2/2) = cubroot(8) = 2.. so here we get a yes answer ..

if we take b = 3 b^2 = 9 cubroot(9/2) is not an integer.. so we get a no answer.

I am finding it difficult to understand how just knowing that a relationship exists is sufficient to answer the question..
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Re: If a and b are positive integer,is (ab)^(1/3) an integer?  [#permalink]

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New post 17 Jan 2017, 21:52
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ajdse22 wrote:
Abhishek009 wrote:
AbdurRakib wrote:
If a and b are positive integer,is \(\sqrt[3]{ab}\) an integer?

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

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

OG Q 2017(Book Question: 300)


FROM STATEMENT - I

We will be unable to comment on \(\sqrt[3]{ab}\) from one variable \(\sqrt{a}\), we need information on the other variable \(b\) to ascertain the value.

FROM STATEMENT - II

This statement is sufficient , because we have the relationship between the two variable...

Hence , correct answer will be Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.


PS : Feel free to revert in case of any doubt..


Thanks ..do you mean the specific relationship or just any relationship in your analysis of statement 2?
If we have the specific relationship which is described above, then shoud we not check by subsitituting in the prompt expression?

If for example I take another relationship ..say a= b/2..
cubroot(ab) = cubroot(b^2/2)
now let us take b = 4 .. b^2 = 16 and cubroot(b^2/2) = cubroot(8) = 2.. so here we get a yes answer ..

if we take b = 3 b^2 = 9 cubroot(9/2) is not an integer.. so we get a no answer.

I am finding it difficult to understand how just knowing that a relationship exists is sufficient to answer the question..


Hi ajdse22,

You have correctly pointed out that not any relation will work. But, the relationship given in statement 2 will work. Let's analyze it.

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

\(\sqrt[3]{ab} = \sqrt[3]{a \sqrt{a}} = \sqrt[3]{a^{3/2}} =\) \(a^{\frac{3}{2} \times \frac{1}{3}\) = \(a^{\frac{1}{2}\) = b

b is an integer; hence definite answer.

Hope this helps.
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Re: If a and b are positive integer,is (ab)^(1/3) an integer?  [#permalink]

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New post 18 Jan 2017, 10:13
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Got this one wrong. On closer scrutiny, statement (2) actually includes the information in statement (1).
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If a and b are positive integer,is (ab)^(1/3) an integer?  [#permalink]

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New post 21 Dec 2017, 12:09
Question : \((ab)^{1/3}\) integer ?

Statement 1: \(\sqrt{a}\) is integer
Let \(\sqrt{a}\) = m
=> \(a = m^2\)
\((ab)^{1/3}\) = (m^2 * b)^ {1/3}
=> if b = m => \((m^2 * m)^ {1/3}\) => \((m^3)^{1/3}\) = m => integer
=> if b not equals m, then \((m^2 * b)^{1/3}\) , may not be integer , say b = 1, then \((m^2)^{1/3}\) => not an integer
=> Insufficient

Statement 2: b = \(\sqrt{a}\)
=> \(b^2 = a\)
=> \((ab)^{1/3}\) = \((b^2 * b)^{1/3}\) => integer => sufficient => Answer (B)
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Re: If a and b are positive integer,is (ab)^(1/3) an integer?  [#permalink]

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Re: If a and b are positive integer,is (ab)^(1/3) an integer?   [#permalink] 05 Jan 2019, 04:36
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