M21-07

Official Solution:

If \(x\), \(y\), and \(z\) are even positive integers, each of the following must be an even integer EXCEPT:

A. \(\frac{xyz}{4}\)
B. \(x^{(y + z)}\)
C. \((x+y)^{z-1}\)
D. \(x-y-z\)
E. \(y^{z-x}\)


Let's evaluate each option:

A. \(\frac{xyz}{4}\)

Given that all the variables are even, the product \(xyz\) will be a multiple of 8, thus \(\frac{xyz}{4}\) will certainly be a multiple of 2, hence even.

B. \(x^{(y + z)}\)

Since all the variables are even and positive, this expression would equate to \(even^{positive \ even \ number}\), which will always be even.

C. \((x+y)^{z-1}\)

Similarly, this expression would equate to \(even^{positive \ odd \ number}\), which will also always be even.

D. \(x-y-z\)

This operation will always yield an even result for even numbers.

E. \(y^{z-x}\)

This expression could yield an even number. However, it's not necessarily even. Remember that any non-zero number raised to the power of 0 is 1. Therefore, if \(z=x\), this option becomes \(y^{z-x}=y^0=1\), which is odd.


Answer: E