M22-33

Official Solution:

If \(x\) and \(y\) are even integers, which of the following must also be an even integer?

A. \(x^y\)
B. \(2\frac{x}{y}\)
C. \(\frac{x - y}{x + y}\)
D. \(\frac{x^2 - y^2}{2}\)
E. \((x + 1)(y - 1)\)


Let's evaluate each option.

A. \(x^y\). If \(y = 0\) and \(x\) is a nonzero even number, then \(x^y\) will be 1, which is odd.

B. \(2*\frac{x}{y}\). If \(x = 2\) and \(y = 4\), then the expression equals 1, which is odd.

C. \(\frac{x - y}{x + y}\). This option might not even be an integer. For example, if \(x=2\) and \(y = 4\), the result is not an integer.

D. \(\frac{x^2 - y^2}{2} = \frac{(x - y)(x + y)}{2} = \frac{even*even}{2} = even*integer = even\). Therefore, this option is always even.

E. \((x + 1)(y - 1)\). Since \(x\) and \(y\) are even integers, \((x + 1)(y - 1)=odd*odd=odd\).

Thus, only option D must always be an even integer.


Answer: D