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Math Expert V
Joined: 02 Sep 2009
Posts: 58142
The sum of m consecutive integers is 8. If the sum of n consecutive in  [#permalink]

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Question Stats: 40% (02:25) correct 60% (01:59) wrong based on 203 sessions

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The sum of m consecutive integers is 8. If the sum of n consecutive integers is m, what is the value of n?

(A) 15
(B) 16
(C) 31
(D) 32
(E) Cannot be determined by the information provided.

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The sum of m consecutive integers is 8. If the sum of n consecutive in  [#permalink]

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If the sum of m consecutive integers is 8, m must be 16.

Starting from -7 to 8 whose sum is 8
The negative integers cancel out the positive integers, except 8, giving us a sum of 8.

Now, the sum of n consecutive integers is 16.
Therefore, n has to be 32(Option D) - range of numbers starting from -15 to 16(15 negative numbers, 1 zero and 16 positive number)
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Intern  B
Joined: 19 Apr 2017
Posts: 32
Location: India
WE: Management Consulting (Consulting)
Re: The sum of m consecutive integers is 8. If the sum of n consecutive in  [#permalink]

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Can someone please explain how you get to "m must be 16"?
Math Expert V
Joined: 02 Aug 2009
Posts: 7831
Re: The sum of m consecutive integers is 8. If the sum of n consecutive in  [#permalink]

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Bunuel wrote:
The sum of m consecutive integers is 8. If the sum of n consecutive integers is m, what is the value of n?

(A) 15
(B) 16
(C) 31
(D) 32
(E) Cannot be determined by the information provided.

Hi..
On why m should be 16 and nothing else..
SUM is 8, and the SUM of any number of consecutive integers will never be 2^x...
example-- if we look at consecutive positive integers it is 1,2,3,4,5,6... Beyond 4, sum of two numbers too goes beyond 8 and other ways to add are 1+2=3, 1+2+3=6, 2+3=5,3+4=7

so it would be 8 itself and that will be possible when numbers are -7,-6,-5,....0,1,2,....6,7,8 here sum will be 8
and number of integers from -7 to 8 are 7 negative integers , 1 zero and 8 positive integers = 7+1+8=16..
so m is 16
again 16 is 2^4, so it will also require 16 as an integer... numbers become -15,-14....0...14,15,16 = 15+1+16=32

D..

only one problem i see if m is 1 that is 1 consecutive integer only 8 then answer does not fit in.
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Re: The sum of m consecutive integers is 8. If the sum of n consecutive in  [#permalink]

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chetan2u wrote:
Bunuel wrote:
The sum of m consecutive integers is 8. If the sum of n consecutive integers is m, what is the value of n?

(A) 15
(B) 16
(C) 31
(D) 32
(E) Cannot be determined by the information provided.

Hi..
On why m should be 16 and nothing else..
SUM is 8, and the SUM of any number of consecutive integers will never be 2^x...
example-- if we look at consecutive positive integers it is 1,2,3,4,5,6... Beyond 4, sum of two numbers too goes beyond 8 and other ways to add are 1+2=3, 1+2+3=6, 2+3=5,3+4=7

so it would be 8 itself and that will be possible when numbers are -7,-6,-5,....0,1,2,....6,7,8 here sum will be 8
and number of integers from -7 to 8 are 7 negative integers , 1 zero and 8 positive integers = 7+1+8=16..
so m is 16
again 16 is 2^4, so it will also require 16 as an integer... numbers become -15,-14....0...14,15,16 = 15+1+16=32

D..

only one problem i see if m is 1 that is 1 consecutive integer only 8 then answer does not fit in.

is there a way to solve this using Arithmatic Progression sum?
Manager  B
Joined: 04 May 2014
Posts: 155
Location: India
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Re: The sum of m consecutive integers is 8. If the sum of n consecutive in  [#permalink]

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if we have odd consecutive integers 3 and 5 than m=2 how do we proceed?
Manager  B
Joined: 18 Mar 2015
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Re: The sum of m consecutive integers is 8. If the sum of n consecutive in  [#permalink]

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pushpitkc wrote:
If the sum of m consecutive integers is 8,
m must be 16.
Starting from -7 to 8 whose sum is 8
The negative integers cancel out the positive integers, except 8 giving us a sum of 8.

Now, sum of n consecutive integers is 16.
n has to be 32(Option D) - range of numbers starting from -15 to 16(15 negative numbers, 1 zero and 16 positive number)

took time to understand this logic ..nice explanation
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Re: The sum of m consecutive integers is 8. If the sum of n consecutive in  [#permalink]

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Bunuel wrote:
The sum of m consecutive integers is 8. If the sum of n consecutive integers is m, what is the value of n?

(A) 15
(B) 16
(C) 31
(D) 32
(E) Cannot be determined by the information provided.

We are given that the sum of m consecutive integers is 8. We see that the m (assuming m > 1) consecutive integers can’t be all positive because we can’t obtain a sum of 8 from m only-positive consecutive integers.. For example, if m = 2, 3 + 4 = 7, and 4 + 5 = 9, and if m = 3, 1 + 2 + 3 = 6, and 2 + 3 + 4 = 9. Thus, some of the integers must be positive and some must be negative, and we see that one of them must, therefore, be 0.

Since the sum is 8, a positive number, there must be more positive integers than negative integers. However, since the negative integers will cancel out their positive counterparts and we have mentioned that there can’t be 2 (or more) consecutive positive integers that add up to 8, there must be exactly 1 more positive integer than the number of negative integers. Furthermore, that positive integer (that won’t cancel with any of its negative counterparts) must be 8. That is, the m consecutive integers must be:

-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8

We see that the sum of all the integers from -7 to 8, inclusive, is 8 since the sum of all the integers from -7 to 7, inclusive, is 0. We also see that there are 16 integers in the above list, so m = 16.

Next we are given that the sum of n consecutive integers is m, which we now know is 16. We can use a similar argument to the one above to conclude that the n (assuming n > 1) consecutive integers can’t be all positive. Furthermore, there must be 1 more positive integer than the number of negative integers and that extra positive integer (the one without a negative counterpart) must be 16. That is, the n consecutive integers must be:

-15, -14, …, -1, 0, 1, …, 14, 15, 16

We see that there are 32 integers in the above list since there are 15 negative integers, 1 zero, and 16 positive integers. Thus, n = 32.

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Re: The sum of m consecutive integers is 8. If the sum of n consecutive in  [#permalink]

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Bunuel wrote:
The sum of m consecutive integers is 8. If the sum of n consecutive integers is m, what is the value of n?

(A) 15
(B) 16
(C) 31
(D) 32
(E) Cannot be determined by the information provided.

Key word = sum of m consecutive integers is 8,

how can we get that sum ??
Sum of 16 integers from -7 to 8, can help us in that

This 16 = m

Now how can we get 16 ??

Sum of 32 integers from -15 to 16 can help us in that

Therefore n = 32
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Intern  B
Joined: 08 Jun 2019
Posts: 3
Re: The sum of m consecutive integers is 8. If the sum of n consecutive in  [#permalink]

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Rule of thumb: Sum of k consecutive nos will be multiple of k, for example, sum of 3 consecutive nos will be multiple of 3.

So, sum of m consecutive nos is 8, so m will be multiple of 8. m can be 16,32...

Now, n is the sum of m consecutive nos, so it will be a multiple of m, i.e, multiple of 16,32..

so if m=16, n=32. Re: The sum of m consecutive integers is 8. If the sum of n consecutive in   [#permalink] 05 Sep 2019, 10:15
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