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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
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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

y is an even number, and sum of even consecutive numbers can't be divided by that even number, so I would go with C

I agree with C. we have x = y.k + y(y-1)/2 in which k is the first element of y consecutive integers. w= z.l + z(z-1)/2 in which k is the first element of z consecutive integers. y=2z so x=2z.k + 2z(2z-1)/2 and x/y = k + (2z-1)/2 cannot be an integer.

I would also go with C. But my explanation is slightly different.

Given , Sum of y consecutive integers is X. Sum of z consecutive integers is W.

Also y = 2z which means we can conclude that y is always even and w can be even or odd.

When I scanned thru the options, I see that options C and D have some relation specified which is nothing but the avg.

If I look at option C which is X/y, it is nothing but the Avg of y consecutive Integers. For even number is consecutive integers, average cannot be an integer since it always is the avg of the 2 middle integers. Hence we can conclude that X / y can never be an integer.

If I loot at option D, which is W/z, again avg of z consecutive integers, can be an integer if z is odd or cannot be an integer if z is even. SO there is a possibility of W/z to be an integer.

Last edited by mrsmarthi on 09 Mar 2009, 05:51, edited 1 time in total.

I agree with C. we have x = y.k + y(y-1)/2 in which k is the first element of y consecutive integers. w= z.l + z(z-1)/2 in which k is the first element of z consecutive integers. y=2z so x=2z.k + 2z(2z-1)/2 and x/y = k + (2z-1)/2 cannot be an integer.

Would you mind explaining the summation formulas that you used?

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

x= sum of y consecutive integers. = (k+1)+(k+2)...+(k+y) (here K can be any integer (+ve or -ve or zero) = yk+ (1+2+3.. y) = yk+ y(y+1)/2

x/y = k+(y+1)/2 = k+(2z+1)/2 --> can't be integer because (2z+1) odd number divided by 2 is always fraction number.
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