If x and y are positive integers and x/y is an even integer, which

Official Solution:

If \(x\) and \(y\) are positive integers and \(\frac{x}{y}\) is an even integer, which of the following can be an odd integer?

A. \(x\)
B. \(xy\)
C. \(x - y\)
D. \(x + 2y\)
E. \(\frac{x}{3}\)


Number picking:

If \(x=2\) and \(y=1\), only option C results in an odd value: \(x-y = 2-1 = 1\), which is odd.

Process of Elimination (POE):

Given \(\frac{x}{y} = \text{even}\), it follows that \(x = y*\text{even} = \text{even}\).

A. \(x\). We've established that \(x\) is even. Discard.

B. \(xy\). Since \(x\) is even, \(xy\) will also be even. Discard.

C. \(x-y\). Keep.

D. \(x + 2y\). The expression \(x + 2y\) evaluates to \(\text{even} + \text{even}*y\), which is always even. Discard.

E. \(\frac{x}{3}\). Dividing an even number by an odd integer either yields an even integer, like \(\frac{6}{3} = 2\), or a non-integer, such as \(\frac{2}{3}\). Discard.

The only option remaining is C, hence it must be correct.


Answer: C