If a is the sum of x consecutive positive integers. b is the

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If a is the sum of x consecutive positive integers. b is the [#permalink]

06 Mar 2012, 00:36

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[Reveal] Spoiler: OA

Last edited by Bunuel on 06 Mar 2012, 03:50, edited 2 times in total.

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Re: sum of x consecutive positive integers [#permalink]

06 Mar 2012, 03:48

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If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?
A. x = 2; y = 6
B. x = 3; y = 6
C. x = 7; y = 9
D. x = 10; y = 4
E. x = 10; y = 7

The sum n consecutive integers is give by: Sum=\frac{(2a_1+n-1)*n}{2} (check Number Theory chapter of Math Book for more: math-number-theory-88376.html);

Notice that:
If n=even=2*odd, so when n (# of consecutive integers) is even but not a multiple 4 then Sum=\frac{(2a_1+n-1)*n}{2}=\frac{(even+even-odd)*(2*odd)}{2}=odd*odd=odd;

If n=even=2*even, so when n is a multiple 4 then Sum=\frac{(2a_1+n-1)*n}{2}=\frac{(even+even-odd)*(2*even)}{2}=odd*even=even;

That's because a set of even number of consecutive integers has half even and half odd terms. The sum of even terms is obviously even. As for odd terms: their sum is even if their number is even (so total # of terms is multiple of 4) and their sum is odd if their number is odd (so total number of terms is even but not a multiple of 4);

So, the sum of 10 (not a multiple of 4) consecutive integers will be odd (the sum of 5 even and 5 odd integers) and the sum of 4 (multiple of 4) consecutive integers will be even (the sum of 2 even and 2 odd integers), so option D is not possible.

Answer: D.

Hope it's clear.
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Manager

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Re: If a is the sum of x consecutive positive integers. b is the [#permalink]

06 Mar 2012, 23:48

Hi,
I am struggling with the explanation.Here is what I had done. but was unable to eliminate answers
the premise was that for x and y to be equal both should be either even or odd
Please take a look and let me know

Attachments

odd even.png [ 6.53 KiB | Viewed 1784 times ]

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Re: If a is the sum of x consecutive positive integers. b is the [#permalink]

07 Mar 2012, 01:46

devinawilliam83 wrote:

I'm not sure what you are trying to say there with the diagram but the 4th row is not correct: if the # of terms is multiple of 4 then the sum is even, regardless of the first term.
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Re: If a is the sum of x consecutive positive integers. b is the [#permalink]

27 Oct 2012, 21:25

For consecutive terms , Sum = median * number of terms. So if m1 and m2 are two medians..

a= m1*x and b = 2*y

as per given condition. a=b => m1*x = m2*y if you put x=10 and y =4

10m1 = 4m2 => 5m1 = 2m2.....

When number of terms are even, the median is always a fraction. I.e 3.5 or 4.5 but when number of terms is odd then median is always an integer.
Now in above case. m1 and m2 both are fractions but m2's fraction can be nullified by 2 which is multiplied by m2. But for m1 it is not the case.

So all combinations are possible but if number of terms are multiple of 4, then the other can not be of the form 4n+1 or 4n+2 or 4n+3.

Hence D
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Re: If a is the sum of x consecutive positive integers. b is the [#permalink]

02 Feb 2013, 11:05

Toooo good Bunuel...
Instead of learning formula i am more comfortable with the conceptual approach. Do you think learning formula is integral for GMAT.

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Re: If a is the sum of x consecutive positive integers. b is the [#permalink]

03 Feb 2013, 03:21

We know that if x=10, and say the first term is odd, then the sum(a) will be

5x(odd+even) = 5x(odd)= odd.

The first term can be even also, nonetheless, the sum will be 5x(even+odd)=5x(odd)= Still odd.

Now for y=4, whatever the first term be,(odd/even) the sum b = 2(odd+even) = 2x(odd)=even.

Thus as odd cant be equal to even, ans is D.

Re: If a is the sum of x consecutive positive integers. b is the [#permalink] 03 Feb 2013, 03:21

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If a is the sum of x consecutive positive integers. b is the

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