runningguy wrote:If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

A) (x+z)/z
B) (y+z)/x
C) (x+y)/z
D) (xy)/z
E) (yz)/x

Note: To avoid confusion, I added some brackets to your answer choice.
Mitch already showed the approach of looking for values of x, y and z, such that we get a non-integer.

Here's another approach:

x is a multiple of z
So, we can say that x = kz (for some integer k)

x is a factor of y
In other words, y is a multiple of x.
So, we can say that y = jx (for some integer j)
IMPORTANT: Since we already know that x = kz, let's replace x with kz to get: y = jkz

So, x = kz, y = jkz and z = z.
Let's plug in these values to see what we get:

A) (x+z)/z = (kz+z)/z = k+1 (= INTEGER)
B) (y+z)/x = (jkz+z)/kz = (jk+1)/k (not necessarily an integer)
C) (x+y)/z = (kz+jkz)/z = k+jk (= INTEGER)
D) (xy)/z = [(kz)(jkz)]/z = jk²z (= INTEGER)
E) (yz)/x = [(jkz)(z)]/kz = jz (= INTEGER)

Answer = B

Cheers,
Brent