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# x and y are positive integers. If the greatest common divisor of x and

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GMAT Club Legend
Joined: 11 Sep 2015
Posts: 4567
GMAT 1: 770 Q49 V46
x and y are positive integers. If the greatest common divisor of x and  [#permalink]

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03 Feb 2017, 06:36
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95% (hard)

Question Stats:

51% (02:24) correct 49% (02:18) wrong based on 199 sessions

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x and y are positive integers. If the greatest common divisor of x and 3y is 9, and the least common multiple of 3x and 9y is 81, then what is the value of 81xy?

A) $$3^5$$

B) $$3^6$$

C) $$3^7$$

D) $$3^8$$

E) $$3^9$$

*Kudos for all correct solutions

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Joined: 11 Sep 2015
Posts: 4567
GMAT 1: 770 Q49 V46
Re: x and y are positive integers. If the greatest common divisor of x and  [#permalink]

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03 Feb 2017, 10:47
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Top Contributor
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GMATPrepNow wrote:
x and y are positive integers. If the greatest common divisor of x and 3y is 9, and the least common multiple of 3x and 9y is 81, then what is the value of 81xy?

A) $$3^5$$

B) $$3^6$$

C) $$3^7$$

D) $$3^8$$

E) $$3^9$$

If the greatest common divisor (GCD) of x and 3y is 9, then the GCD of 3x and 9y is 27
Given: the least common multiple (LCM) of 3x and 9y is 81

Nice rule: (GCD of j and k)(LCM of j and k) = jk
So, (27)(81) = (3x)(9y)
Rewrite as: (3^3)(3^4) = 27xy

Our goal is to determine the value of 81xy.
So, take (3^3)(3^4) = 27xy and multiply both sides by 3 to get: (3)(3^3)(3^4) = (3)(27xy)

Simplify: 3^8 = 81xy

Cheers,
Brent
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Re: x and y are positive integers. If the greatest common divisor of x and  [#permalink]

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03 Feb 2017, 06:56
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6
GMATPrepNow wrote:
x and y are positive integers. If the greatest common divisor of x and 3y is 9, and the least common multiple of 3x and 9y is 81, then what is the value of 81xy?

A) $$3^5$$

B) $$3^6$$

C) $$3^7$$

D) $$3^8$$

E) $$3^9$$

*Kudos for all correct solutions

GCD of x & 3y= 9
LCM of 3x & 9y =81
thus LCM of x & 3y= 27
Number = LCM * GCD
x*3y= 27*9 = 3^5
xy= 3^4

so 81xy = 3^4*3^4 = 3^8

Ans D
##### General Discussion
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Joined: 17 May 2015
Posts: 238
Re: x and y are positive integers. If the greatest common divisor of x and  [#permalink]

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03 Feb 2017, 06:55
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2
GMATPrepNow wrote:
x and y are positive integers. If the greatest common divisor of x and 3y is 9, and the least common multiple of 3x and 9y is 81, then what is the value of 81xy?

A) $$3^5$$

B) $$3^6$$

C) $$3^7$$

D) $$3^8$$

E) $$3^9$$

*Kudos for all correct solutions

GCD of x and 3y is 9.

Let x = 9a and 3y = 9b or y = 3b, where a and b are co-prime numbers.

LCM of 3x and 9y = 81(given) LCM(3x,9y) = LCM(3*9a, 9*3b) = 27ab = 81 ==> ab =3

81xy = 81*9a*3b = 81*9*3*3 = $$3^{8}$$
GMAT Club Legend
Joined: 11 Sep 2015
Posts: 4567
GMAT 1: 770 Q49 V46
x and y are positive integers. If the greatest common divisor of x and  [#permalink]

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03 Feb 2017, 11:00
Top Contributor
GMATPrepNow wrote:
x and y are positive integers. If the greatest common divisor of x and 3y is 9, and the least common multiple of 3x and 9y is 81, then what is the value of 81xy?

A) $$3^5$$

B) $$3^6$$

C) $$3^7$$

D) $$3^8$$

E) $$3^9$$

When it comes to solving integer properties questions, it's often useful to be able to come up with values that meet the given conditions.
For example, if we're told that positive integers j and k have a greatest common divisor of 15, what are some possible values of j and k? Can you quickly come up with 3 or 4 pairs of values?
Some possibilities are: j = 15 and k = 15 (easy!), or j = 30 and k = 15, or j = 30 and k = 45, or j = 15 and k = 150, etc.

What about the given question? Can you find values of x and y such that the greatest common divisor of x and 3y is 9, and the least common multiple of 3x and 9y is 81?
How about x = 27 and y = 3?
Or perhaps x = 9 and y = 9?

Once we're able to identify values that satisfy the given information, it's easy to determine the value of 81xy

If we use x = 27 and y = 3, then 81xy = (81)(27)(3) = (3^4)(3^3)(3^1) = 3^8
If we use x = 9 and y = 9, then 81xy = (81)(9)(9) = (3^4)(3^2)(3^2) = 3^8

Cheers,
Brent
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x and y are positive integers. If the greatest common divisor of x and  [#permalink]

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03 Feb 2017, 18:04
1
1
GMATPrepNow wrote:
x and y are positive integers. If the greatest common divisor of x and 3y is 9, and the least common multiple of 3x and 9y is 81, then what is the value of 81xy?

A) $$3^5$$

B) $$3^6$$

C) $$3^7$$

D) $$3^8$$

E) $$3^9$$

*Kudos for all correct solutions

if gcd of x and 3y=9,
and lcm of 3x and 9y=81,
and problem does not require unique values for x and y,
then value for both x and y can be 9
81xy=81*9^2=6561=3^8
D
Intern
Joined: 23 Jan 2018
Posts: 12
Re: x and y are positive integers. If the greatest common divisor of x and  [#permalink]

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23 Mar 2018, 18:34
1
GMATPrepNow wrote:
x and y are positive integers. If the greatest common divisor of x and 3y is 9, and the least common multiple of 3x and 9y is 81, then what is the value of 81xy?

A) $$3^5$$

B) $$3^6$$

C) $$3^7$$

D) $$3^8$$

E) $$3^9$$

*Kudos for all correct solutions

I figured that X could be 9 given the greatest common divisor of x and 3y is 9, and it follows that if least common multiple of 3x (27) and 9y is 81 then y could be 9, from there 81xy = 3^4 * 3^2 * 3^2 = 3^8 Answer choice D
Intern
Joined: 15 Jul 2017
Posts: 3
x and y are positive integers. If the greatest common divisor of x and  [#permalink]

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17 Feb 2019, 02:22
Why waste time with unnecessary calculations if this question can be solved in 20 seconds, bearing in mind Euclid's algorithm.

Make it x=9 and y=9, so that: x=9 and 3y=27

Then plug in these values in 81xy, so that 81*9*9=6561 and infer that 6561=3^8

You can see that converting 6561 into 3^8 takes the most time with this question.

ecobiz
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Posts: 763
Location: India
Concentration: Nonprofit, Sustainability
Schools: ISB
GMAT 1: 630 Q47 V30
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Re: x and y are positive integers. If the greatest common divisor of x and  [#permalink]

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28 Feb 2020, 16:54
Let (a,b) be co-primes then,

x=9a &
3y=9b OR y=3b

To find:- 81 * 27 * (a*b)

Now 3x and 9y is basically 27a and 27b both have a multiple 81 (LCM), so we can say (a,b) can be (1,3) or (3,1)

Now: 81 * 27 * (a*b) is = 3^4 * 3^3 * (3) = 3^8

GMATPrepNow wrote:
x and y are positive integers. If the greatest common divisor of x and 3y is 9, and the least common multiple of 3x and 9y is 81, then what is the value of 81xy?

A) $$3^5$$

B) $$3^6$$

C) $$3^7$$

D) $$3^8$$

E) $$3^9$$

*Kudos for all correct solutions
Re: x and y are positive integers. If the greatest common divisor of x and   [#permalink] 28 Feb 2020, 16:54
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