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If a and b are both positive integers such that a < b, which of the fo
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07 Jul 2018, 08:25
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If a and b are both positive integers such that a < b, which of the fo
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07 Jul 2018, 09:15
Bunuel wrote: 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) \(a<\sqrt{ab}<\sqrt{b}\) \(a=\sqrt{aa}\) and b=\(\sqrt{bb}\)since a<b...... \(\sqrt{aa}<\sqrt{ab}<\sqrt{bb}.......................a<\sqrt{ab}<b\) A
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1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
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Re: If a and b are both positive integers such that a < b, which of the fo
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07 Jul 2018, 09:35
Bunuel wrote: 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) \(a<\sqrt{ab}<\sqrt{b}\) 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.
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Re: If a and b are both positive integers such that a < b, which of the fo
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25 Jul 2018, 14:48
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?



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If a and b are both positive integers such that a < b, which of the fo
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25 Jul 2018, 19:17
rohithtv89 wrote: 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? 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.
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Re: If a and b are both positive integers such that a < b, which of the fo
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25 Jul 2018, 19:48
Hi PKN. Yes. I didn't read it correctly. Thanks for pointing that out!



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Re: If a and b are both positive integers such that a < b, which of the fo
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25 Jul 2018, 20:04
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.
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25 Jul 2018, 23:34
Orionstar wrote: 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 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))
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Re: If a and b are both positive integers such that a < b, which of the fo
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26 Jul 2018, 15:25
Bunuel wrote: 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) \(a<\sqrt{ab}<\sqrt{b}\) If we let a = 4 and b = 9, we see that only A is true, since √(4 x 9) = √36 = 6. Answer: A
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Re: If a and b are both positive integers such that a < b, which of the fo
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27 Jul 2018, 03:12
PKN wrote: Bunuel wrote: 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) \(a<\sqrt{ab}<\sqrt{b}\) 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. 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<b so what`s wring with it ? thanks and have a great day !:)



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Re: If a and b are both positive integers such that a < b, which of the fo
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27 Jul 2018, 03:31
dave13 wrote: PKN wrote: Bunuel wrote: 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) \(a<\sqrt{ab}<\sqrt{b}\) 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. 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<b so what`s wring with it ? thanks and have a great day !:) 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.
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Re: If a and b are both positive integers such that a < b, which of the fo
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27 Jul 2018, 07:46
Hi PKN
In this question, is there a chance that SQRT(A*B) is negetive?



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Re: If a and b are both positive integers such that a < b, which of the fo
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27 Jul 2018, 09:15
ETLim wrote: Hi PKN
In this question, is there a chance that SQRT(A*B) is negetive? \(\sqrt{}\) 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 \(\sqrt[4]{x}\), 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.
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If a and b are both positive integers such that a < b, which of the fo
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27 Jul 2018, 09:28
ETLim wrote: Hi PKN
In this question, is there a chance that SQRT(A*B) is negetive? 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.
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If a and b are both positive integers such that a < b, which of the fo
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28 Jul 2018, 04:35
Bunuel wrote: 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) \(\sqrt{ab}<a<b\)
(E) \(a<\sqrt{ab}<\sqrt{b}\) 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) \(1<\sqrt{2}<\sqrt{2}\)............................................... EliminateAnswer: A



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If a and b are both positive integers such that a < b, which of the fo
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28 Jul 2018, 04:43
Hi PKNCan you please explain the highlighted text in chetan2u 's approach? Quote: If a and b are both positive integers such that a < b, which of the following must be true? Quote: \(a=\sqrt{aa}\) and b=\(\sqrt{bb}\)since a<b...... \(\sqrt{aa}<\sqrt{ab}<\sqrt{bb}\)......................\(.a<\sqrt{ab}<b\) I could not proceed from \(\sqrt{a}\) \(\sqrt{b}\) = \(\sqrt{ab}\)
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If a and b are both positive integers such that a < b, which of the fo
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28 Jul 2018, 05:53
adkikani wrote: Hi PKNCan you please explain the highlighted text in chetan2u 's approach? Quote: If a and b are both positive integers such that a < b, which of the following must be true? Quote: \(a=\sqrt{aa}\) and b=\(\sqrt{bb}\)since a<b...... \(\sqrt{aa}<\sqrt{ab}<\sqrt{bb}\)......................\(.a<\sqrt{ab}<b\) I could not proceed from \(\sqrt{a}\) \(\sqrt{b}\) = \(\sqrt{ab}\) 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 a<b, (c) I am sure we can write \(a=\sqrt{a}*\sqrt{a}\) & \(b=\sqrt{b}*\sqrt{b}\) Substituting in (c), we have \(\sqrt{a}*\sqrt{a} < \sqrt{b}*\sqrt{b}\)(d) Again, a<b (You know we can multiply positive numbers on both sides of the inequality)(Given a and b are positive integers) Or, \(a*b<b^2\) (multiplying 'b' both sides) Or, \(\sqrt{ab}<\sqrt{b^2}\) (we can take square root on both sides of inequalities when they are positive) Or, \(\sqrt{ab}< b\)(e) Similarly, b>a (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 \(a<\sqrt{ab}<b\)We have derived it.
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If a and b are both positive integers such that a < b, which of the fo &nbs
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