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.
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