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

Question

If a and b are positive integer, is  \sqrt {ab}  an integer?

(1)  \sqrt{a}  is an integer

(2)  b=\sqrt{a}

OG Q 2017(Book Question: 300)

Bunuel · Accepted Answer

If a and b are positive integers, is  \sqrt {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 {ab}=\sqrt {b^2*b}=b=integer . Sufficient.

Answer: B.

Bunuel · Answer

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.

14101992 · Answer

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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Neeraj91 · Answer

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!

AmritaSarkar89 · Answer

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.

Mike2805 · Answer

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?

CrackverbalGMAT · Answer

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.

Cheers,
Crackverbal Academics Team

Kritisood · Answer

Hi  VeritasKarishma   Bunuel

I have a doubt:

the question appears as below in the online version of the Quant Review. I thought under the cube root it was a^b and hence marked E. I'm scared now that I'm misreading the question and marking the wrong answer to a question that I know the concept for. pls, advise!!

Ahmed9955 · Answer

Hey guys,

Please help me identify where i am wrong.?  Bunuel

if I solve by substituting the value of b in given equation.

(ab)^1/3 = a^1/3 * b^1/3 
=a*1/3 * a^1/6 
= a^3/6 = a^1/2

now we dont know whether a^1/2 is an integer or not

by combining S1 and S2 i know a^1/2 is an integer, hence answer c

KarishmaB · Answer

Don't worry. Now that I look at it closely, it could be mistaken for a^b. That won't happen in the actual GMAT.

errorlogger · Answer

Your approach is absolutely correct. But the only mistake that you have done here is by overlooking the question stem.
It clearly states that b is an integer.
And as per statement 2:
b = a^1/2
So, a^1/2 has to be an integer and hence A

woohoo921 · Answer

CrackverbalGMAT

The official explanation has ³√(ab) =³√(a√a) =³√(√a^3)= √a=b

I realize that the approach Bunuel took above in using b instead to substitute was easier, but I am very confused on substituting a in to solve based on the algebra here.

If I were to solve using exponents instead of the cube/square root signs above, is this right?
(ab)^(1/3)
((a)*(a)^.5)^(1/3)
(a)^(3/2)*(1/3)
=a^.5=squareroot(a)

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

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