Bunuel wrote:

The operation ? is deﬁned for all integers x and y as \(x?y = xy - y\). If x and y are positive integers, which of the following CANNOT be zero?

A. \(x?y\)

B. \(y?x\)

C. \((x-1)?y\)

D. \((x+1)?y\)

E. \(x?(y-1)\)

Let’s analyze each answer choice.

A) x?y

x?y = 0

xy - y = 0

y(x - 1) = 0

y = 0 or x = 1

Since y can’t be 0, then x = 1. However, as long as x = 1, regardless of what y is, we will have x?y = 0. Thus, A is not the answer.

B) y?x

y?x = 0

yx - x = 0

x(y - 1) = 0

x = 0 or y = 1

Since x can’t be 0, then y = 1. However, as long as y = 1, regardless of what x is, we will have y?x = 0. Thus, B is not the answer.

C) (x - 1)?y

(x - 1)?y = 0

(x - 1)y - y = 0

xy - 2y = 0

y(x - 2) = 0

y = 0 or x = 2

Since y can’t be 0, then x = 2. However, as long as x = 2, regardless of what y is, we will have (x - 1)?y = 0. Thus C is not the answer.

D) (x + 1)?y

(x + 1)?y = 0

(x + 1)y - y = 0

xy = 0

y = 0 or x = 0

We see that the only way (x + 1)?y can be 0 is if y = 0 or x = 0. However, we are given that both x and y are positive integers. Therefore, there is no way (x + 1)?y can be 0. Thus, D is the answer.

Answer: D

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Jeffery Miller

Head of GMAT Instruction

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