This is testing a very particular rule in consecutive integers: if you have an EVEN number of terms, the sum will never be divisible by the number of terms. If you have an ODD number of terms, the sum *will* be divisible by the number of terms.
This is because, by definition, SUM = (AVERAGE)*(NUMBER OF TERMS), and in a consecutive set, AVERAGE = MEDIAN.
If you have an EVEN number of terms, the median is a non-integer:
[4, 5, 6, 7]
median = 5.5
sum = (5.5)(4) = 22
If we divide the sum by the number of terms, we get a non-integer: the median. Thus, the sum is not divisible by the number of terms.
If you have an ODD number of terms, the median is an integer:
[4, 5, 6, 7, 8]
median = 6
sum = (6)(5) = 30
If we divide the sum by the number of terms, we get an integer: the median. Thus, the sum is divisible by the number of terms.
So in this problem, if y = 2z, then y must be even: there are an EVEN number of terms in the set. The sum cannot possibly be divisible by the number of terms in an even set, so x cannot be divisible by y.
The answer is C.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
