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If x, y and z are different positive integers,

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If x, y and z are different positive integers, [#permalink] New post 22 May 2018, 06:56
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Question Stats:

80% (00:52) correct 20% (00:35) wrong based on 10 sessions
If x, y and z are different positive integers, and y is the greatest common divisor of x, y and z, which of the following MUST be true?

i) y < x
ii) the greatest common divisor of x and z is greater than y
iii) the greatest common divisor of x and y is y

A) ii only
B) i and ii only
C) i and iii only
D) ii and iii only
E) I, ii and iii
[Reveal] Spoiler: OA

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Re: If x, y and z are different positive integers, [#permalink] New post 22 May 2018, 08:02
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This is a must be true question hence the correct ans choice must be true at all times.

Let us pick three +ve numbers 3,6,12
Since y is the greatest common divisor it is safe to take 3 as y

This makes y=3< 6 or 12 hence whatever number is picked for x option i. is correct
If y = 3 x and y can interchangeably be 6 and 12 for which the HCF is 6 option ii. is also correct
Since Y is the least number in the three series and it divides both x and z; The HCF of y,z or y,x is always x option iii. is also correct

However,

Instead of taking three number set of a multiple of a number such as the one we have taken earlier which is a multiple of 3. If we take numbers such as 1,2,3
option ii. will not be correct.

Only i. and iii. are correct all the time
option C
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This is my response to the question and may be incorrect. Feel free to rectify any mistakes

GRE Instructor
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Joined: 10 Apr 2015
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Re: If x, y and z are different positive integers, [#permalink] New post 26 May 2018, 05:16
Expert's post
GreenlightTestPrep wrote:
If x, y and z are different positive integers, and y is the greatest common divisor of x, y and z, which of the following MUST be true?

i) y < x
ii) the greatest common divisor of x and z is greater than y
iii) the greatest common divisor of x and y is y

A) ii only
B) i and ii only
C) i and iii only
D) ii and iii only
E) I, ii and iii


Let's examine each statement separately...

i) y < x
If y is the greatest common divisor of x, y and z, we can also say that y is a divisor of x
IMPORTANT CONCEPT: The divisors of a number are always less than or equal to that number. For example, the divisors of 10 are {1, 2, 5, 10).
So, the divisors of x must be less than or equal to x
If y is a divisor of x, then y must be less than or equal to x
Since we're told that x, y and z are different positive integers, we can be certain that y is less than x
So, statement i is TRUE
Check the answer choices....ELIMINATE A and D

ii) the greatest common divisor of x and z is greater than y
Let's test some values of x, y and z that satisfy the given information ( x, y and z are different positive integers, and y is the greatest common divisor of x, y and z)
x = 4, y = 2 and z = 6. Notice that y (2) is the greatest common divisor of 4, 2 and 6
Here, the greatest common divisor of x (4) and z (6) is 2.
So, the greatest common divisor of x and z is NOT greater than y
So, statement ii is FALSE
Check the answer choices....ELIMINATE B and E

By the process of elimination (and without having to even look at statement iii), the correct answer must be C

Cheers,
Brent
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Re: If x, y and z are different positive integers,   [#permalink] 26 May 2018, 05:16
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If x, y and z are different positive integers,

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