The greatest common divisor of x and 3y

[Brent@GMATPrepNow]

Here's a 700-level question I just created.

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

Answer: D

[DavidG@VeritasPrep]

Here's a 700-level question I just created.

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

Answer: D

Brent original!

If the greatest common divisor of x and 3y is 9, or 3^2, it means that 3^2 represents the overlap between x and 3y. So we know x = 3^2 * some integer. And 3y already has a 3, so y = 3 * some integer

If the least common multiple of 3x and 9y is 81, or 3^4, it means that either 3x or 9y contains 3^4, and that neither number exceeds this value. If x = 3^3, then 3x = 3^4. y could still be 3.

So now we have a scenario that will satisfy both conditions. If x = 3^3 and y = 3, then the GCD of x and 3y would be 3^2, and the LCM of 3x and 9y would be 3^4.

Now plug x = 3^3 and y = 3 into 81*xy = 81* 3^3 * 3 = 3^4 * 3^3 * 3 = 3^8. The answer is D

[DavidG@VeritasPrep]

(Note that another scenario will work. If, say, x = 3^2 and y = 3^2, the GCD of x and 3y would be 3^2 and the LCM of 3x and 9y would be 81.

But the result will be the same: 81*x * y = 81 * 3^2 * 3^2 = 3^4 * 3^2 * 3^2 = 3^8.

[Brent@GMATPrepNow]

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

Answer: D

[Brent@GMATPrepNow]

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

[GMATGuruNY]

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

Since the GCD of x and 3y is 9, 9 must divide into both x and 3y.
Thus, x≥9 and y≥3.

We can PLUG IN THE ANSWERS, which represent the value of 81xy = 3�xy.

A: 3�xy = 3�, implying that xy = 3
Since x must be greater than or equal to 9, eliminate A.

B: 3�xy = 3�, implying that xy = 3² = 9
If x=9, then y=1.
Since y must be greater than or equal to 3, eliminate B.

C: 3�xy = 3�, implying that xy = 3³ = 27
Test whether x=9 and y=3 satisfy the constraint that the LCM of 3x and 9y is 81.
If x=9, then 3x = 27.
If y=3, then 9y = 27.
The LCM of 27 and 27 is not 81 but 27.
Eliminate C.

D: 3�xy = 3�, implying that xy = 3� = 81
If x=9 and y=9, then the GCD of x=9 and 3y=27 is 9.
Test whether x=9 and y=9 satisfy the constraint that the LCM of 3x and 9y is 81.
If x=9, then 3x = 27.
If y=9, then 9y = 81.
The LCM of 27 and 81 is 81.
Success!

The correct answer is D.

[shashank.ism]

Here's a 700-level question I just created.

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

Answer: D

LCM of 3x and 9y is 81 = 3^4
hence LCM of x and 3y is 3^3 = 27

Also HCF od x and 3y = 9

Also LCM * HCF = product of numbers
-> (27)*(9) = (x)*(3y)
-> 3^5 = 3xy
-> 3^8 = 81xy

Hence D