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

Question

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

rohit8865 · Accepted Answer

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

BrentGMATPrepNow · Answer

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

Answer: D

Cheers,
Brent

ganand · Answer

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}

BrentGMATPrepNow · Answer

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

gracie · Answer

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

hawesg · Answer

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

ecobiz · Answer

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

LeoN88 · Answer

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

BrushMyQuant · Answer

Given that LCM of 3x and 9y is 81 and GCD of x and 3y is 9 and we need to find the value of 81xy

Lets solve the Problem using Two Methods:

Method 1:

LCM of 3x and 9y = 81

If we divide 3x and 9y by 3 to get x and 3y then their LCM = (LCM of 3x and 9y) / 3 =  \frac{81}{3}  = 27

LCM of two numbers * GCD of two numbers = Product of the two numbers

=> LCM(x,3y) * GCD(x,3y) = x * 3y = 3xy
=> 3xy = 27*9
=> 27*3xy = 27 * 27 * 9 =  3^8 
=> 81xy =  3^8

So,  Answer will be D

Method 2:

GCD of x and 3y = 9

If we multiply x and 3y by 3 to get 3x and 9y then their GCD = (GCD of x and 3y) * 3 = 27 * 3 = 81

LCM of two numbers * GCD of two numbers = Product of the two numbers

=> LCM(3x,9y) * GCD(3x,9y) = 3x * 9y = 27xy
=> 27xy = 81*81
=> 3*27xy = 3*81*81 =  3^8 
=> 81xy =  3^8

So,  Answer will be D 
Hope it helps!

To learn more about LCM and GCD watch the following videos

https://www.youtube.com/watch?v=7jPV6JD0908

https://www.youtube.com/watch?v=vtSsAjcF-0Y

Dereno · Answer

The greatest common divisor of x and 3y is 9.

x = 9a

3y = 9b

Then, y = 3b

The least common multiple (LCM) of 3x and 9y = 81

3x = 3* 9a = 27a

9y = 9*3b = 27b

LCM ( 3x, 9y) = 3*9 * x*y = 27*a*b = 81

Thus,  a*b = 3

The Value of 81xy

=  81 * 9a * 3b

= 81 * 9 * 3 * a * b

= 81 * 9 * 3 * 3

= 9*9*9*3*3

=  3^8

Option D

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

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