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Re: If X is the GCD of three positive integers whose LCM is Y,then which
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28 Nov 2016, 12:45

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stonecold wrote:

If X is the GCD (greatest common divisor) of three positive integers x,y,z whose LCM is Y,then which of the following statements must be true.

A)\(X*Y=x*y*z\) B)Least possible value of X is z C)x,y,z are all divisible by Y D)x,y, and z are all divisible by X E)X must be less than z

IMPORTANT: If d is a divisor of N, we can also say that N is divisible by d For example, since 6 is a divisor of 24, we can also say that 24 is divisible by 6

If X is the greatest common divisor (GCD) of x, y, and z, then we know that X is a divisor of x, X is a divisor of y, and X is a divisor of z So, by the above property, we can say that x, y, and z are all divisible by X Answer:

Re: If X is the GCD of three positive integers whose LCM is Y,then which
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28 Nov 2016, 12:54

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stonecold wrote:

If X is the GCD of three positive integers x,y,z whose LCM is Y,then which of the following statements must be true.

A) XY = xyz B) Least possible value of X is z C) x,y,z are all divisible by Y D) x,y,z are all divisible by X E) X must be less than z

In my above post, I showed why D is the correct answer. We can also try showing why the other answer choices are incorrect.

A) XY = xyz If x = 2, y = 2 and z = 2, then X = 2 and Y = 2. In this case XY = (2)(2) = 4, and xyz = (2)(2)(2) = 8 So, we can see that XY ≠ xyz So, this statement need not be true.

B) Least possible value of X is z If x = 3, y = 2 and z = 2, then X = 1 and Y = 6. In this case, X is less than z So, this statement need not be true.

C) x,y,z are all divisible by Y If x = 3, y = 2 and z = 2, then X = 1 and Y = 6. In this case, x, y and z are NOT divisible by Y (e.g., 3 is not divisible by 6) So, this statement need not be true.

E) X must be less than z If x = 2, y = 2 and z = 2, then X = 2 and Y = 2. In this case XY = (2)(2) = 4, and xyz = (2)(2)(2) = 8 So, we can see that X is NOT less than z So, this statement need not be true.

By the process of elimination, we can see that the correct answer is D.

Re: If X is the GCD of three positive integers whose LCM is Y,then which
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13 Jul 2018, 06:54

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