Problem Solving

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

Answer: C
Difficulty level: 600 - 650
Source: www.gmatprepnow.com

Brent@GMATPrepNow 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

Answer: C
Difficulty level: 600 - 650
Source: www.gmatprepnow.com

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