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x is the sum of y consecutive integers. w is the sum of z

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

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New post 18 Nov 2009, 21:37
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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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Re: integers  [#permalink]

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New post 24 Dec 2010, 07:49
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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

Can someone please prove?

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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New post 19 Nov 2009, 04:53
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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.

Answer : C
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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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New post 18 Nov 2009, 23:37
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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=5, TRUE

Case 2: x > w
if y = [2,3] z = [1]
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 = [2] w = 2
2/1 is an integer, TRUE

Case 5: x/z is an integer
if x = 9, z = [2]
9/1 is an integer, TRUE

Am I doing something wrong here?
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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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New post Updated on: 19 Nov 2009, 05:28
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=5, TRUE

Case 2: x > w
if y = [2,3] z = [1]
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 = [2] w = 2
2/1 is an integer, TRUE

Case 5: x/z is an integer
if x = 9, z = [2]
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, 04:07.
Last edited by kp1811 on 19 Nov 2009, 05:28, edited 1 time in total.
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Re: integers  [#permalink]

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New post 16 Dec 2009, 03:08
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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
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Re: integers  [#permalink]

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New post 16 Dec 2009, 03:11
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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
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Re: Intresting problem  [#permalink]

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New post 02 Jan 2010, 14:08
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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
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Re: integers  [#permalink]

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New post 31 Jan 2010, 13:02
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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.

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

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New post 21 Apr 2011, 03:08
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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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Re: x is the sum of y consecutive integers. w is the sum of z consecutive  [#permalink]

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New post 21 Apr 2011, 03:19
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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New post 21 Apr 2011, 03:29
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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New post 21 Apr 2011, 03:55
1
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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New post 21 Apr 2011, 04:12
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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New post 21 Apr 2011, 04:36
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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New post 21 Apr 2011, 05:17
Yup.. Even i got C.. :-D
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!! :?: :?:
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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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New post 21 Apr 2011, 06:09
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.. :-D
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!! :?: :?:
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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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New post 22 Apr 2011, 06:34
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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New post Updated on: 22 Apr 2011, 11:06
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, 09:37.
Last edited by gmatpapa on 22 Apr 2011, 11:06, edited 1 time in total.
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Re: Someone please explain this one  [#permalink]

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New post 11 May 2012, 03:36
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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Re: Someone please explain this one &nbs [#permalink] 11 May 2012, 03:36

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