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Re: x and y are positive integers. If the greatest common diviso [#permalink]
GreenlightTestPrep wrote:


If the greatest common divisor (GCD) of x and 3y is 9, then the GCD of 3x and 9y is 27



Is there an algebraic proof for the above?
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Re: x and y are positive integers. If the greatest common diviso [#permalink]
1
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



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Re: x and y are positive integers. If the greatest common diviso [#permalink]
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Re: x and y are positive integers. If the greatest common diviso [#permalink]
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