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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] New post 05 Mar 2012, 23:36
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 06 Mar 2012, 02:50, edited 2 times in total.
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Re: sum of x consecutive positive integers [#permalink] New post 06 Mar 2012, 02:48
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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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Re: If a is the sum of x consecutive positive integers. b is the [#permalink] New post 06 Mar 2012, 22:48
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
odd even.png [ 6.53 KiB | Viewed 3417 times ]

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Re: If a is the sum of x consecutive positive integers. b is the [#permalink] New post 07 Mar 2012, 00:46
Expert's post
devinawilliam83 wrote:
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


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] New post 27 Oct 2012, 20:25
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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] New post 02 Feb 2013, 10:05
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] New post 03 Feb 2013, 02:21
Expert's post
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.
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Re: sum of x consecutive positive integers [#permalink] New post 30 Nov 2013, 06:43
Bunuel wrote:
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.



Brunel ,

Sum = n/2( 2a +(n-1)d ).....i still not able to get ur sum formula . I guess for consecutive number you are taking d=1 ...
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Re: sum of x consecutive positive integers [#permalink] New post 01 Dec 2013, 05:20
Expert's post
archit wrote:
Bunuel wrote:
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.



Brunel ,

Sum = n/2( 2a +(n-1)d ).....i still not able to get ur sum formula . I guess for consecutive number you are taking d=1 ...


Yes, the common difference between the numbers in a set of consecutive numbers is 1.
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Re: If a is the sum of x consecutive positive integers. b is the [#permalink] New post 01 Dec 2013, 07:09
devinawilliam83 wrote:
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



this is a very bad question ... no way a GMAT question... very time consuming ...
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Re: If a is the sum of x consecutive positive integers. b is the [#permalink] New post 11 Apr 2014, 03:53
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devinawilliam83 wrote:
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



Sum of 2 consecutive integers = Odd
Sum of 4 consecutive integers = Even
Sum of 6 consecutive integers = Odd
Sum of 8 consecutive integer = Even and so on is the pattern.

If we look at answer option D. Sum of 10 consecutive integer = Odd, Sum of 4 consecutive integers = Even.

Odd cannot be equal to Even.

Hence the answer is D.
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