blaster wrote:If y ≠ 3 and 2x/y is a prime integer greater than 2, which of the following must be true?

I. x = y
II. y = 1
III. x and y are prime integers.

(A) None
(B) I only
(C) II only
(D) III only
(E) I and II

(I) If x = y = 1, then 2x/y = 2, not greater than 2.
(II) y =1 implies 2x/y = 2x
Put x = 1 so 2x = 2, again not greater than 2.
(III) If x = 3, y = 5 then 2x/y = 6/5, not greater than 2, and not a prime integer.

The correct answer is [spoiler](A)[/spoiler].

Rahul@gurome wrote:

blaster wrote:If y ≠ 3 and 2x/y is a prime integer greater than 2, which of the following must be true?

I. x = y
II. y = 1
III. x and y are prime integers.

(A) None
(B) I only
(C) II only
(D) III only
(E) I and II

(I) If x = y = 1, then 2x/y = 2, not greater than 2.
(II) y =1 implies 2x/y = 2x
Put x = 1 so 2x = 2, again not greater than 2.
(III) If x = 3, y = 5 then 2x/y = 6/5, not greater than 2, and not a prime integer.

I'm concerned that this response might be confusing to some test takers. Here, we do not want to assume that I, II and III are true, and then try to determine if 2x/y is a prime greater than two as is done above; that's logically backwards. We know that 2x/y is a prime greater than 2 -- that's an incontrovertible fact -- and we need to determine whether I, II or III is always true. It's just coincidental that the above gives the right answer.

You can see why the approach above is backwards by imagining a very straightforward question like:

If x=3, what must be true?

(I) x is prime

Obviously (I) must be true. If, however, you assume (I) is true and test whether x = 3 is true, you'll find it's not true for most values of x that you might choose. You'd get the wrong answer because you'd be working in the wrong direction.

To do the question below in the logically correct direction:

If y ≠ 3 and 2x/y is a prime integer greater than 2, which of the following must be true?

I. x = y
II. y = 1
III. x and y are prime integers.

we need to begin with values of x and y for which 2x/y is an odd prime (observing the restriction that y is not 3). Well, our numbers may be x = 6, and y = 4, in which case none of I, II or III is true.

this_time_i_will wrote:IMO D.
For III: we need 2x/y to be an integer that must be prime. For, 2x/y to be integer, y must properly divide 2 or x.If y divdes x then 2x/y can not be prime. so y should divide 2. This means y = 2. Now if 2x/y is a prime and y=2, this gives x as prime.
so both x and y sud be prime.

I'd just add that there's an issue with the logic here. For 2x/y to be an integer, then y must divide 2x. That does not mean that y is a proper divisor of either 2 or of x alone. If x is odd, y might be equal to 2f, where f is any factor of x; then y will not be a divisor of 2 or of x, but will be a divisor of 2x.

That might be easier to see with numbers: if x = 15, then 2x = 30. Let y=6. Then y is a divisor of 2x, but is not a divisor of 2 or of x alone.