x and y are positive integers such that x

Question

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

KarishmaB · Accepted Answer

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

Answer (E)

EMPOWERgmatRichC · Answer

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: E

GMAT assassins aren't born, they're made,
Rich

cherryli2015 · Answer

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

chetan2u · Answer

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

chetan2u · Answer

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...

EMPOWERgmatRichC · Answer

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

mvictor · Answer

x is smaller than y

x*sqrt(y) = 6*sqrt(6)
sqrt  = sqrt(36*6)
sqrt  = 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.

RastogiSarthak99 · Answer

I can see why 36 is the choice for wrong answers mostly. So here's how I did it:

Remember x and y are positive integers. and x is less than or y is greater than x.

we know that x^2 * y = 216

If we prime factorize 216 we get 3^3 * 2^3

Basically you take one of the 2's and give it to y, so Y = 3^3 * 2 = 54
and x^2 remains with 2^2 and thus x = 2.

xy = 54 * 2 = 108

you could take one of the 3's and do it the other way but that won't give any of the options mentioned. (72)

ThatDudeKnows · Answer

Square both sides:
 x^2y = 216

Let's just try a few values.
x = 1, y = 216, xy = 216 ... not an answer choice.
x = 2, y = 216/4 = 54, xy = 108 ... Yay!
That is ALL we need to do. No reason to get all fancy.

Answer choice E.

.
.
.

Bonus:
x = 3, y = 216/9 = 24, xy = 72 ... not an answer choice.
x = 4, y = 216/16 = 13.something ... invalid since y is not an integer.
x = 5, y = 216/25 = 8.something ... invalid since y is not an integer.
x = 6, y = 216/36 = 6 ... invalid since x is not less than y.
If we keep going x will be greater than y. We didn't need to test all of these, but even if we'd had to, it wasn't that bad. Brute force and trial-and-error are frequently (usually) faster than figuring out formulas.

saynchalk · Answer

Easiest way to do it

Here, X is either 2 or 3, and it becomes 6 from either 9 or 4 becoming square rooted from y (and becoming 3 or 2).

Therefore the answers are either 72 or 108. 108 is the only one listed hence E is the answer.

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

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