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# If X is the GCD of three positive integers whose LCM is Y,then which

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Schools: Boston U '20 (M)
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If X is the GCD of three positive integers whose LCM is Y,then which  [#permalink]

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28 Nov 2016, 11:15
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63% (01:41) correct 37% (01:11) wrong based on 183 sessions

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

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Re: If X is the GCD of three positive integers whose LCM is Y,then which  [#permalink]

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28 Nov 2016, 11:45
Top Contributor
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

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Brent
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Re: If X is the GCD of three positive integers whose LCM is Y,then which  [#permalink]

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28 Nov 2016, 11:54
Top Contributor
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.

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Current Student
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Re: If X is the GCD of three positive integers whose LCM is Y,then which  [#permalink]

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28 Apr 2017, 21:31
Firstly ->
GCD*LCM=product of integer is applicable to only a pair of 2 integers.
So incase of 3 integers this property does not hold true.

Secondly -> All the integers must be divisible by the GCD.
Hence D is always true.

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Re: If X is the GCD of three positive integers whose LCM is Y,then which  [#permalink]

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01 May 2017, 20:31
x,y,z are all divisible by X
GCD is the greatest common divisor hence x,y,z must be divisible by X
its not a 700 level question
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Re: If X is the GCD of three positive integers whose LCM is Y,then which  [#permalink]

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13 Jul 2018, 05:54
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Re: If X is the GCD of three positive integers whose LCM is Y,then which &nbs [#permalink] 13 Jul 2018, 05:54
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