Data Sufficiency

If x and y are positive integers, is x a multiple of y?

(1) 2x is a multiple of y.
(2) 2(y^2) + y = 2x

Official answer = B

ziyuenlau wrote:If x and y are positive integers, is x a multiple of y?

(1) 2x is a multiple of y.
(2) 2(y^2) + y = 2x

Statement 1:
Case 1: x=1, 2x=2 and y=1, with the result that 2x is divisible by y
In this case, x is a multiple of y.
Case 2: x=1, 2x=2 and y=2, with the result that 2x is divisible by y
In this case, x is NOT a multiple of y.
INSUFFICIENT.

Statement 2:
RULE:
If x and k are positive integers, then x and kx+1 are COPRIMES: they share no factors other than 1.

2y² + y = 2x
y(2y+1) = 2x
(y/2)(2y+1) = x.

x is the product of two factors:
y/2 and 2y+1.

Since x must be an integer, the factor on the left -- y/2 -- implies that y is EVEN.
In accordance with the rule above, y and 2y+1 are coprimes: they SHARE NO FACTORS other than 1.
Since y is greater than y/2, y cannot divide into y/2.
Since y shares no factors with 2y+1, and y cannot divide into y/2, y cannot divide into (y/2)(2y+1).
Thus:
y cannot divide evenly into x, implying that x is not multiple of y.
SUFFICIENT.

The correct answer is B.