arashyazdiha wrote: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
A)x = w
B)x > w
C)x/y is an integer
D)w/z is an integer
E)x/z is an integer
OA after some discussions
Let n = the smallest integer.
Since y=2z, we know that y is even.
x/y is the average of the y integers.
The average of consecutive integers = (biggest + smallest)/2.
Consider the following cases:
y = 2 consecutive integers:
Smallest = n, biggest = n+1.
x/y = average = (biggest + smallest)/2 = ((n+1) + n)/2 = (2n+1)/2 = n + 1/2.
x/y is not an integer.
y = 4 consecutive integers.
Smallest = n, biggest = n+3.
x/y = average = (biggest + smallest)/2 = ((n+3) + n)/2 = (2n+3)/2 = n + 3/2.
x/y is not an integer.
y = 6 consecutive integers:
Smallest = n, biggest = n+5.
x/y = average = (biggest + smallest)/2 = ((n+5) + n)/2 = (2n+5)/2 = n + 5/2.
x/y is not an integer.
Do you see the pattern?
x/y = n + (y-1)/2.
Since y is even, y-1 must be odd.
Thus, x/y -- the average of the even number of integers -- cannot be an integer.
The correct answer is
C
Here's the take-away:
The average of an even number of consecutive integers is NOT an integer.Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
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