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# If Z1, Z2, Z3 .....Zn is a series of consecutive positive

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Manager
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If Z1, Z2, Z3 .....Zn is a series of consecutive positive [#permalink]  13 Jul 2004, 10:27
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If Z1, Z2, Z3 .....Zn is a series of consecutive positive integers, is the sum of the all integers in this series odd?

(1) ( Z1, Z2, Z3 .....Zn ) / n is an odd integer

(2) n is an odd integer.

Director
Joined: 01 Feb 2003
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(assuming (Z1,Z2,Z3....Zn) = Z1+Z2+Z3+.....+Zn

Difficulty level: easy (2 on a scale of 1 to 10)
Intern
Joined: 18 Jun 2004
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Evaluating choice 1

1) if choice 1 is true then numerator and denominator are odd.

odd/odd = odd (15/5,45/9, etc)
only exception is n/n = 1 which is not possible in this case since numeration is sum of n numbers

!!Sufficient!!
2) choice 2
Let assume n = 3(odd) (1+2+3)=6

!!!insufficient!!

Hence A

I have not practiced enought to give diff. level. Although, it took me about 3+ min to figure out the ans.
Director
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Re: DS: series of consecutive positive integers [#permalink]  13 Jul 2004, 13:29
1 Min
A
Not sure what diff level this prob is.
It's either easy or medium
Senior Manager
Joined: 21 Mar 2004
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pbajaria wrote:
Evaluating choice 1

1) if choice 1 is true then numerator and denominator are odd.

odd/odd = odd (15/5,45/9, etc)
only exception is n/n = 1 which is not possible in this case since numeration is sum of n numbers

!!Sufficient!!

pbajaria,
I just wanted to know how you concluded that both Numberator and Denominator is odd from your explanation above.

6/2 = 3
here Nr and Dr are both even and the quotient is odd.

Now , here's how i concluded that n is odd.

Given that Expression E = Sum(Z1...Zn)/n = odd

Sum(Z1...Zn) = n[Z1+Zn]/2
Now,
E = [Z1+Zn]/2 = odd

therefore Z1+Zn = 2*odd = even
from this we know that both Z1 and Zn are either odd or even.

If n is even Z1 and Zn will never be both odd or both even.
Only if n is odd, Z1 and Zn are both odd or both even and then only can their sum be even.

So the series can only be one of the following types
1 2 3
2 3 4 etc
i.e. n has to be odd. A is sufficient.
You can pick numbers and solve it faster, but if you know the concept
you have an upper hand, just in case picking numbers doesn't workout sometime.

B.
In this case I picked numbers.
given n is odd.
let the series be 123. sum = 6
let the series be 234. sum = 9
So B is insufficient.

What I generally do is like this. If I have to prove something is always true then I try to use the algebraic and arithmetic concepts.

When I dont have to prove that something is always true, I pick numbers.

- ash
_________________

ash
________________________
I'm crossing the bridge.........

CIO
Joined: 09 Mar 2003
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no body has mentioned the most important thing:

the average of any list of consecutive numbers is always the one in the middle.

If the list has an even number of numbers, then there is no middle, and it's the average of the two middle numbers, which will always be something and a half. So if the sum of the list divided by the number of numbers is an integer, there must be an odd number of numbers.

And if it's odd, then the middle number is odd, and the sum would be an odd number times an odd number, and hence an odd number. No need for algebra, just a great concept.

For more, see: http://www.integratedlearning.net/city_la_ian_video55.shtml

number 2 is insufficient since just because there is an odd number of numbers, doesn't tell us anything about the middle - it could be odd or even.

All in all, I think this is a medium question. I think that for most people who don't have the kind of intense gmat experience that many of the posters here have, this would be a difficult question.
Manager
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OA is A. Good job!

Manager
Joined: 21 Jun 2004
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Re: DS: series of consecutive positive integers [#permalink]  15 Jul 2004, 22:32
afife76 wrote:
If Z1, Z2, Z3 .....Zn is a series of consecutive positive integers, is the sum of the all integers in this series odd?

(1) ( Z1, Z2, Z3 .....Zn ) / n is an odd integer

(2) n is an odd integer.

From the bsic pricipals of odd and evn

Even/ odd = even

odd/odd = odd So

If sum (Z1, z2, .... Zn) / n = odd means, sum is ODD

Re: DS: series of consecutive positive integers   [#permalink] 15 Jul 2004, 22:32
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