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 < xIf 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 TRUECheck the answer choices....ELIMINATE A and D

ii) the greatest common divisor of x and z is greater than yLet'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 FALSECheck 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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Brent Hanneson – Creator of greenlighttestprep.com

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