x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

1. x = w
2. x > w
3. x/y is an integer
4. w/z is an integer
5. x/z is an integer

Explanation given:
For any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms. For example, the sum of 1, 2, and 3 (three consecutives -- an odd number) is 6, which is a multiple of 3. For any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms. For example, the sum of 1, 2, 3, and 4 (four consecutives -- an even number) is 10, which is not a multiple of 4.

The question tells us that y = 2z, which allows us to deduce that y is even. Since y is even, then the sum of y integers, x, cannot be a multiple of y. Therefore, x/y cannot be an integer; choice C is the correct answer.


Question:
Can you please explain another way to solve this problem? Is there another way to do it instead of plugging in numbers (which would take a long time in this case)?

y = 2z
if z = 1, y 2
if z = 2, y = 4
if z = 3, y = 6
if z = 4, y = 8

so y is always even and average of even consecutive integers is always not an integer.

lets write down some even consecutive integers. suppose y = 4:

so the integers are: a, a+1, a+2, a+3
x = a + a+1 + a+2 + a+3
x = 4a + 6
so 4a+6 is not evenly divisible by 4. therefore x/y is not an integer...

wow i forgot to put my username.

UPA's solution is akin to how I would solve it as well. I usually find it helpful to express consecutive sets using variables like a, a+1, a+2, etc. and seeing what insights I can draw from that analysis.

Rey