This problem relies on knowledge a special property about the sum of consecutive integers. For any evenly-spaced set (like consecutive integers), the sum is equal to the median times the number of terms. For example, take the set 3 + 4 + 5 + 6 + 7. You could just add them up, or you could take the median, 5, and multiply it by the number of terms. 5 x 5 = 25.

When you have an odd number of terms in the set, your median will be an integer. So, if you divide the sum by the number of terms, you get an integer - the median. In other words, the sum is divisible by the number of terms. If you have an even number of terms, though (like 3 + 4 + 5 + 6), the median is not an integer (in this case it's 4.5). So when you divide the sum by the number of terms, you get a non-integer, and thus the sum is not divisible by the number of terms.

Odd number of terms... the sum is divisible by the number of terms.
Even number of terms... the sum is not divisible by the number of terms.

In this problem, we're told that x is the sum of y consecutive integers. We're also told that y = 2z. Therefore, y has to be even, so the sum, x, will not be divisible by the number of terms, y.