If a and b are both positive integers such that a

Question

If a and b are both positive integers such that a < b, which of the following must be true?

(A)  a < \sqrt{ab} <b

(B)  \sqrt{a} < \sqrt{ab} < \sqrt{b}

(C)  \sqrt{a} < \sqrt{b} < \sqrt{ab}

(D)  \sqrt{ab}<a<b

(D)  a<\sqrt{ab}<\sqrt{b}

chetan2u · Answer

a=\sqrt{aa}  and b= \sqrt{bb} since a<b......  \sqrt{aa}<\sqrt{ab}<\sqrt{bb}.......................a<\sqrt{ab}<b

A

PKN · Answer

Let a=4 & b=16 (a<b)
(A)  a < \sqrt{ab} <b : 4<8<16   (True)  
(B)  \sqrt{a} < \sqrt{ab} < \sqrt{b} : 2<8<4 (False)
(C)  \sqrt{a} < \sqrt{b} < \sqrt{ab} : 2<4<8 (False)
(D)  \sqrt{ab}<a<b : 8<4<16 (False)
(E)  a<\sqrt{ab}<\sqrt{b} : 2<8<4 (False)

Ans. (A)

P.S.- Perfect square are chosen for checking purpose since square roots are involved in the answer choices.

rohithtv89 · Answer

If I choose a = 1 and b = 2 (both positive integers), then sqrt(b) = sqrt(ab).
So no option "must be true". Did I miss anything?

PKN · Answer

Hi  rohithtv89 ,
Let's check options A to E with your chosen (a,b), i.e. (1,2)

(A)  a < \sqrt{ab} <b :  1<\sqrt{2}=1.41(approx.)<2    (True)  
(B)  \sqrt{a} < \sqrt{ab} < \sqrt{b} :  1<\sqrt{2}<\sqrt{2}  (False)
(C)  \sqrt{a} < \sqrt{b} < \sqrt{ab} :  1<\sqrt{2}<\sqrt{2}  (False)
(D)  \sqrt{ab}<a<b :  \sqrt{2}<1<2  (False)
(E)  a<\sqrt{ab}<\sqrt{b} :  1<\sqrt{2}<\sqrt{2} (False)

You might have checked option A with  \sqrt{b}  but it is 'b' only.

rohithtv89 · Answer

Hi  PKN . Yes. I didn't read it correctly. Thanks for pointing that out!

Orionstar · Answer

Why option C is not correct. If I take a as 4 and b as 9.
Than √4=2 less than √9=3 less than √4*9=6.

What I am missing here.

Posted from my mobile device

PKN · Answer

Hi  Orionstar ,
First of all, this is a   MUST BE TRUE   question. We have to find out the correct relationship among a,b,  \sqrt{a} , \sqrt{b} , and  \sqrt{ab}  in terms of magnitude. And that relationship  MUST HOLD TRUE    at all the possible positive integer values of a and b, available in the universe  . (with the given limitation:- a<b)

With the chosen pairing (4,9), the relationship in C is true. Is the given relationship in option 'C" true for all positive integer values of a and b? Answer is No (Say a=1 , b=2, (a<b))

JeffTargetTestPrep · Answer

If we let a = 4 and b = 9, we see that only A is true, since √(4 x 9) = √36 = 6.

Answer: A

dave13 · Answer

PKN hey there, can you please explain how can option C be wrong. 2<4<8 if A=2 B= 4 o correct 2<4 or a ? thanks and have a great day !:-)

PKN · Answer

Hi  dave13  ,
Have a great day!!

First of all, this is a MUST BE TRUE question. We have to find out the correct relationship among a,b, a√a,b√b, and ab−−√ab in terms of magnitude. And that relationship MUST HOLD TRUE at all the possible positive integer values of a and b, available in the universe. (with the given limitation:- a<b)

With the chosen pairing (2,4), the relationship in C is true( 1.41<2<2.82 ). Is the given relationship in option 'C" true for all positive integer values of a and b? Answer is No (Say a=1 , b=2, (a<b))

The correct answer option or relationship must hold true for all positive integers a and b (a<b), not on a piecemeal basis:-), true in some cases and false in other cases.  Hence, options C is discarded.

Thanking You.

ETLim · Answer

Hi PKN

In this question, is there a chance that SQRT(A*B) is negetive?

Bunuel · Answer

\sqrt{}  denotes a function. Mathematically the square root function cannot give negative result. When the GMAT provides the square root sign for an even root, such as  \sqrt{x}  or  \sqrt {x} , then the only accepted answer is the positive root.

That is,  \sqrt{25}=5 , NOT +5 or -5. In contrast, the equation  x^2=25  has TWO solutions, +5 and -5.  Even roots have only a positive value on the GMAT.

OFFICIAL GUIDE:
 \sqrt{n}  denotes  the positive number  whose square is n.

PKN · Answer

Hi  ETLim ,

Since it is mentioned in the question stem that 'a' and 'b' are positive integers, hence multiplication of two positive integers will always yield a positive integer. Therefore, a*b=positive. Hence,  \sqrt{ab}=\sqrt{{positive}}=Positive  in GMAT.

Mo2men · Answer

Let's think to DISAPPROVE the choices

a=1 & b=2

(A)  a < \sqrt{ab} <b .............(A)  1 < \sqrt{2} <2 ............. Keep

(B)  \sqrt{1} < \sqrt{2} < \sqrt{2} ................. Eliminate

(C)  \sqrt{1} < \sqrt{2} < \sqrt{2} ................. Eliminate

(D)  \sqrt{2}<1<2 .................................................. Eliminate

(E)  1<\sqrt{2}<\sqrt{2} ............................................... Eliminate

Answer: A

adkikani · Answer

Hi  PKN

Can you please explain the highlighted text in  chetan2u  's approach?

I could not proceed from  \sqrt{a}   \sqrt{b}  =  \sqrt{ab}

PKN · Answer

Hi adkikani , First of all, you must be knowing the following property of exponent:- (a^m)*(a^n)=a^{m+n} Let m=n=\frac{1}{2} , now a^{\frac{1}{2}} * a^{\frac{1}{2}}=a^{\frac{1}{2}+\frac{1}{2}} =a----(1) Also, you know, a^{1/2}=\sqrt{a} -------(b) Now moving to the question:- Given aa (You know we can multiply positive numbers on both sides of the inequality)(Given a and b are positive integers) Or, b*a>a^2 (multiplying 'a' both sides) Or, \sqrt{ab}>\sqrt{a^2} (we can take square root on both sides of inequalities when they are positive) Or, \sqrt{ab}> a ------------------(f) Combining (d),(e), and (f), we have a<\sqrt{ab}

achloes · Answer

Hi Jeff, these were the numbers that I picked as well, only to realize that C also holds true.

√a = √4 = 2
√b = √9 = 3 
√ab = √(4x9) = √36 = 6

2 < 3 < 6

I solved the issue by testing a = 1 and b = 100 for the two remaining options.

How do I determine which numbers are best to test in such cases?

HoudaSR · Answer

I solved this algebraically as following:

GAURI2828 · Answer

I took a=1 and b=2 it worked in this case but is it okay to consider it? Can involvement of 1 maybe cause issue ?

If a and b are both positive integers such that a < b, which of the fo

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