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x and y are positive integers. If the greatest common diviso
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16 Apr 2019, 08:30
16 Apr 2019, 08:30
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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) \(3^9\)
A) \(3^5\)
B) \(3^6\)
C) \(3^7\)
D) \(3^8\)
E) \(3^9\)
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: x and y are positive integers. If the greatest common diviso
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17 Apr 2019, 05:27
17 Apr 2019, 05:27
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GreenlightTestPrep 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) \(3^9\)
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
Cheers,
Brent
General Discussion
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: x and y are positive integers. If the greatest common diviso
[#permalink]
17 Apr 2019, 05:28
17 Apr 2019, 05:28
1
GreenlightTestPrep 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) \(3^9\)
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
