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Manager  Joined: 11 Aug 2009
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x is the sum of y consecutive integers. w is the sum of z co  [#permalink]

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Question Stats: 36% (02:25) correct 64% (02:26) wrong based on 738 sessions

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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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Joined: 02 Sep 2009
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nonameee 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

Quote:
For any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms.
But For any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms.

• If $$k$$ is odd, the sum of $$k$$ consecutive integers is always divisible by $$k$$. Given $$\{9,10,11\}$$, we have $$k=3$$ consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.

• If $$k$$ is even, the sum of $$k$$ consecutive integers is never divisible by $$k$$. Given $$\{9,10,11,12\}$$, we have $$k=4$$ consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

• The product of $$k$$ consecutive integers is always divisible by $$k!$$, so by $$k$$ too. Given $$k=4$$ consecutive integers: $$\{3,4,5,6\}$$. The product of 3*4*5*6 is 360, which is divisible by 4!=24.
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Re: x is the sum of y consecutive integers. w is the sum of z co  [#permalink]

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For a set of 'n' consecutive integers, the sum of integers is :
(a) always a multiple of 'n' when n is ODD
(b) never a multiple of 'n' when n is EVEN
(Note : Try it out!)

From y = 2z, we can conclude that x is the sum of consecutive integers for an EVEN number of terms since y will always be even.

Thus x/y can never be an integer.

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Re: x is the sum of y consecutive integers. w is the sum of z co  [#permalink]

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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

Case 1: x = w
if y = [2,3] z = 
5=5, TRUE

Case 2: x > w
if y = [2,3] z = 
5>1, TRUE

Case 3: x/y is an integer
if y = [2,3,4] x = 9
9/3 is an integer, TRUE

Case 4: w/z is an integer
if z =  w = 2
2/1 is an integer, TRUE

Case 5: x/z is an integer
if x = 9, z = 
9/1 is an integer, TRUE

Am I doing something wrong here?
Manager  Joined: 30 Aug 2009
Posts: 211
Location: India
Concentration: General Management
Re: x is the sum of y consecutive integers. w is the sum of z co  [#permalink]

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h2polo 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

Case 1: x = w
if y = [2,3] z = 
5=5, TRUE

Case 2: x > w
if y = [2,3] z = 
5>1, TRUE

Case 3: x/y is an integer
if y = [2,3,4] x = 9
9/3 is an integer, TRUE

Case 4: w/z is an integer
if z =  w = 2
2/1 is an integer, TRUE

Case 5: x/z is an integer
if x = 9, z = 
9/1 is an integer, TRUE

Am I doing something wrong here?

well well well...I came to 3 taking y as a multiple of 2 (considering y=2z and y,z both are integers)
yet I may be wrong...

Originally posted by kp1811 on 19 Nov 2009, 03:07.
Last edited by kp1811 on 19 Nov 2009, 04:28, edited 1 time in total.
Manager  Joined: 30 Aug 2009
Posts: 211
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delta09 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

C

a) x= w if x = (1+2+3+4+5+6) = 21 and w = (6+ 7+8)=21.
b) x>w if x= (1+2+3+4+5+6) = 21 and w = (4+5+6)= 15
c) x/y => x = (1+2+3+4+5+6) = 21 and y = 6 then x/y is NOT an integer
=> x = (1+2+3+4) and y = 4 then x/y is NOT an integer
d) w/z =>w = (6+ 7+8)=21 and z=3 then w/z = 3
e) x/z => x = (1+2+3+4+5+6) = 21 and z = 3 then x/z = 7
Manager  Joined: 09 May 2009
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delta09 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

IMO (C).

y will always be even since y=2z

assume 6 numbers
n-2 n-1 n n+1 n+2 n+3
x=6n+3, y=6, z=3

x/z = 2n+1

x/y = n+1/2 ==> cannot be integer
Intern  Joined: 18 Jul 2009
Posts: 24

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OA is C

sum of even no is not always odd.

consider 1+2+3+4=10 which is even

For any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms.

consider above example 10 is not multiple of 4.

But For any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms.

1+2+3=6 multiple of 3
Manager  Joined: 22 Dec 2009
Posts: 225

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delta09 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

since y = 2z... it is even...

x is the sum of y (even no) of integers....

Hence x/y would never be an integer....

Therefore C....

Note
For any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms.
But For any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms.
Manager  Joined: 26 Mar 2007
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x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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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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Posts: 52
Location: Bangalore India
Schools: LBS, HBS, ISB, Kelloggs, INSEAD
Re: x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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kannn 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

Why the answer is not A for this one, as if y consecutive integers are 1 2 3 4 5 and z consecutive integers are 1 2 3 4 5 6 7 8 9 10 then x(sum of y ints) not equal to y(sum of z ints).

please someone explain, whats the flaw in my opinion?
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Re: x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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atulmogha wrote:
kannn 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

Why the answer is not A for this one, as if y consecutive integers are 1 2 3 4 5 and z consecutive integers are 1 2 3 4 5 6 7 8 9 10 then x(sum of y ints) not equal to y(sum of z ints).

please someone explain, whats the flaw in my opinion?

You should try and find a case where the statement (a) will hold true or try and find the option which must not be true. Question asks: "each of the following could be true EXCEPT".

c) x/y -- integer. or x/even -- integer; thus x will have to be even to make the statement true but as we know "x" is the sum of equal numbers of odds and evens and will always be odd. This statement "Must not" be true or always be false.
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Re: x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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fluke wrote:
atulmogha wrote:
kannn 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

Why the answer is not A for this one, as if y consecutive integers are 1 2 3 4 5 and z consecutive integers are 1 2 3 4 5 6 7 8 9 10 then x(sum of y ints) not equal to y(sum of z ints).

please someone explain, whats the flaw in my opinion?

You should try and find a case where the statement (a) will hold true or try and find the option which must not be true. Question asks: "each of the following could be true EXCEPT".

c) x/y -- integer. or x/even -- integer; thus x will have to be even to make the statement true but
as we know "x" is the sum of equal numbers of odds and evens and will always be odd
. This statement "Must not" be true or always be false.

Sume of 1,2,3, 4 (even number of integers) = 10, which is even. Am I missing something? Help please.
Though I agree that it is not divisible by 4 i.e.m 2y.
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Re: x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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kannn wrote:
fluke wrote:

Sume of 1,2,3, 4 (even number of integers) = 10, which is even. Am I missing something? Help please.
Though I agree that it is not divisible by 4 i.e.m 2y.

No, you are right. Thanks for correcting me. Rule is as follows:

1. If n is odd, the sum of "n" consecutive integers is always divisible by n.
2. If n is even, the sum of "n" consecutive integers is never divisible by n.
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Re: x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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This is heavily cloaked problem masking the issue that -
The average of even number of integers is never an integer.

Here x = sum of even number of consecutive integers ( y).
y = 2z (even)
x / y = fraction. Answer C
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Re: x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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Yup.. Even i got C.. But I'm still confused as to y E is not the OA.. E states: x/z is an integer. Same logic for could be true works here?? HELP!!  Director  Status: Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
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Re: x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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Lets assume that x is the sum of first y consecutive integers
x = y(y+1)/2 ----------(1)
y = 2z
Substituting the value of y in (1)
x = 2z(2z+1)/2 = z(2z+1)
x/z = 2z+1. We know that z is an integer.

Hence x/z can be an integer.

varunmaheshwari wrote:
Yup.. Even i got C.. But I'm still confused as to y E is not the OA.. E states: x/z is an integer. Same logic for could be true works here?? HELP!!  Manager  Status: It's "Go" Time.......
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Re: x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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The answer must be "C" because if there are n consecutive integers & n is even then mean of those integer will never be an integer..
say 2,3,4,5 -> (2+3+4+5)/4 = 14/4 = 3.5

Easy one got it in 57 sec.
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Re: x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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y= even number.
x= sum of y numbers

Rule is that average of even number of consecutive integers is never even. Therefore, x/y will never be an integer.

Do we need "w" here? or its just there to confuse? Because anyways its given z is a positive integer.
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Originally posted by gmatpapa on 22 Apr 2011, 08:37.
Last edited by gmatpapa on 22 Apr 2011, 10:06, edited 1 time in total.
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The question can be restated as:
If x is the sum of y consecutive integers, and w is the sum of y/2 consecutive integers, and neither of y or z is 0, then which of the following can NEVER be true?

Note that since y>0 and z=y/2 and z>0 and both y and z are integers, y will always be an even integer.

(a) x=w. This is certainly possible. Take x=-2,-1,0,1,2,3 and w=0,1,2
(b) x>w. This is certainly possible. Take x=-2,-1,0,1,2,3 and w=-2,-1,0
(c) Correct answer. You can never make a case that satisfies this. Why?
Sum of the AP = (y/2) [2a+(y-1)*1]
If y = 2k as y is always even, Sum = x = k [2a+ (2k-1)]
If we divide this by y, i.e. 2k, we get x/y = k[2a+2k-1]/2k. This is an odd number divided by an even number, and so can never be an integer.
(d) This is certainly possible. Evaluates to [2a+(z-1)]/2, which can be true if z is odd.
(e) This is certainly possible. Take x=-2,-1,0,1 and then z=2, and x/z=-1, which is an integer.
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