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# x and y are positive integers such that x < y. If

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Joined: 07 Aug 2011
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GMAT 1: 630 Q49 V27
x and y are positive integers such that x < y. If  [#permalink]

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Updated on: 16 Feb 2015, 03:59
2
21
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Difficulty:

65% (hard)

Question Stats:

60% (02:26) correct 40% (02:08) wrong based on 265 sessions

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x and y are positive integers such that x < y. If $$x\sqrt{y} = 6\sqrt{6}$$, then xy could equal

A. 36
B. 48
C. 54
D. 96
E. 108

Originally posted by Lucky2783 on 15 Feb 2015, 23:24.
Last edited by Bunuel on 16 Feb 2015, 03:59, edited 1 time in total.
Renamed the topic and edited the question.
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Re: x and y are positive integers such that x < y. If  [#permalink]

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15 Feb 2015, 23:42
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Lucky2783 wrote:
x and y are positive integers such that x < y. If x sqrt(y) = 6 sqrt(6) , then xy could equal

36
48
54
96
108

First instinct, when you see
$$x*\sqrt{y} = 6*\sqrt{6}$$, you say that x = 6, y = 6 will satisfy this. But note that x < y.

So y should be 6*Perfect square so that when you square root it, you are left with $$\sqrt{6}$$.

Try y = 6*4. Then x will be 3 so that you get 3*2 = 6 outside the square root.
x*y = 3*24 = 72 (not in the options)

Try y = 6*9. Then x will be 2 so that you get 2*3 = 6 outside the square root.
x*y = 2*54 = 108

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Re: x and y are positive integers such that x < y. If  [#permalink]

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16 Feb 2015, 19:38
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HI Lucky2783,

This question is a mix of Exponent rules and Number Properties.

We're told that X and Y are POSITIVE INTEGERS and that X < Y. We're also told that X(sqrt Y) = 6(sqrt 6).

For example:
(sqrt 50) = 5(sqrt 2)

Here, we have to go "in reverse" and put the 6 "back in" to the radical...

6(sqrt 6) = (sqrt 216)

The question asks for a possible value of XY...

We have (X)(X)(Y) = 216 as a reference

Since the answers are ALL integers, we're looking for one, that when multiplied by another positive integer, gives us 216... Notice how 108 is exactly HALF of 216....?

IF...
(X)(Y) = 108 and X=2, then (X)(X)(Y) = 216.

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# Rich Cohen

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ Intern Joined: 15 Feb 2015 Posts: 13 Re: x and y are positive integers such that x < y. If [#permalink] ### Show Tags 16 Feb 2015, 19:50 1 X and Y are positive integer, which eliminates all negative situation. x\sqrt{y} = 6\sqrt{6} -> \sqrt{x*x*y} = \sqrt{6*6*6} 6*6*6 = 3*2*3*2*3*2 since x<y -> x= 2, xy = 3*2*3*2*3 = 108 Math Expert Joined: 02 Aug 2009 Posts: 7831 Re: x and y are positive integers such that x < y. If [#permalink] ### Show Tags 16 Feb 2015, 20:02 1 2 Lucky2783 wrote: x and y are positive integers such that x < y. If $$x\sqrt{y} = 6\sqrt{6}$$, then xy could equal A. 36 B. 48 C. 54 D. 96 E. 108 hi, since it is given $$x\sqrt{y} = 6\sqrt{6}$$... It is clear that $$\sqrt{y} = t\sqrt{6}$$... and t can be 2 or 3 or 6 as 6=2*3.. 1)let t=3 so x will be 2.. $$\sqrt{y} = 3\sqrt{6}$$... so y=$$(3\sqrt{6})^2$$ y=54 and xy=108.. ans E.. although we already have our ans, the two other possible values can be.. 2) the other possible value is t=2 and x=3 xy=3*$$(2\sqrt{6})^2$$=3*24=72.. not a choice 3) the other possible value is t=6 and x=1 xy=1*$$(6\sqrt{6})^2$$=1*216=216.. not a choice _________________ Math Expert Joined: 02 Aug 2009 Posts: 7831 x and y are positive integers such that x < y. If [#permalink] ### Show Tags 16 Feb 2015, 20:08 EMPOWERgmatRichC wrote: HI Lucky2783, This question is a mix of Exponent rules and Number Properties. We're told that X and Y are POSITIVE INTEGERS and that X < Y. We're also told that X(sqrt Y) = 6(sqrt 6). In most cases, we're asked to "simplify" a radical... For example: (sqrt 50) = 5(sqrt 2) Here, we have to go "in reverse" and put the 6 "back in" to the radical... 6(sqrt 6) = (sqrt 216) The question asks for a possible value of XY... We have (X)(X)(Y) = 216 as a reference Since the answers are ALL integers, we're looking for one, that when multiplied by another positive integer, gives us 216... Notice how 108 is exactly HALF of 216....? IF... (X)(Y) = 108 and X=2, then (X)(X)(Y) = 216. Final Answer: GMAT assassins aren't born, they're made, Rich hi, the ans is correct but there are other values in choices which when multiplied by integer would give you 216.. A. 36.... 36*6=216 C. 54....54*4=216 E. 108...108*2=216... so it is important to see which of these satisfies the equation x<y to get the correct answer... _________________ EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 15026 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: x and y are positive integers such that x < y. If [#permalink] ### Show Tags 16 Feb 2015, 20:19 Hi chetan2u, As noted in my explanation, X and Y are both INTEGERS and X < Y; we need an answer that is in the format (X)(X)(Y) = 216 Of the 3 values that you listed, 2 of them do NOT fit that pattern. 36(6) = 216, but you would end up with (6)(6)(6) which is NOT a match (since X is NOT less than Y). 54(4) = 216, but you would end up with either (4)(4)(13.5) or (root54)(root54)(4), NEITHER of which is a match (since they both include NON-integer values). 108(2) = 216, which gives us (2)(2)(54), which IS the correct answer. GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com *****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***** # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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x and y are positive integers such that x < y. If  [#permalink]

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14 Dec 2015, 19:57
x is smaller than y

$$x*sqrt(y) = 6*sqrt(6) sqrt[(x^2)*y] = sqrt(36*6) sqrt[(x^2)*y] = sqrt(216)$$

ok, let's find prime factorization of 216
ok, 216 = 2 * 108
108 = 2 * 54
stop right here!
we have two 2's, and 54. since x^2 * y = 216, it might be the case that x=2 and y = 54. 2*54 = 108, and it is in the answer choices. since it is a "could be" question, there is no need to check further.
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Re: x and y are positive integers such that x < y. If  [#permalink]

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25 Nov 2018, 09:51
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Re: x and y are positive integers such that x < y. If   [#permalink] 25 Nov 2018, 09:51
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