Bunuel wrote:

If \(X\), \(Y\), and \(Z\) are positive integers, is \((X-Y) * (Y-Z) * (X-Z) \gt 0\)?

(1) \(X^2 + YZ = XY + XZ\)

(2) \(XY - Y^2 = XZ - YZ\)

Here's what I did to solve this:

(X-Y) * (Y-Z) * (X-Z) > 0

i.e. neither (X-Y) nor (Y-Z) nor (X-Z) =0

or X!=Y and Y!=Z and X!=Z

(1) \(X^2 + YZ = XY + XZ\)

or \(X^2 - XZ = XY - YZ\)

or \(X(X - Z) = Y(X - Z)\)

or \(X = Y\)

Hence the (X-Y) * (Y-Z) * (X-Z) = 0 * (Y-Z) * (X-Z) = 0.

Sufficient.

(2) \(XY - Y^2 = XZ - YZ\)

or \(XY - XZ = Y^2 - YZ\)

or \(X(Y - Z) = Y(Y - Z)\)

or \(X = Y\)

Hence the (X-Y) * (Y-Z) * (X-Z) = 0 * (Y-Z) * (X-Z) = 0.

Sufficient.

Thus D is the Answer