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The mean of a set of n consecutive positive integers is m1. One of the

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The mean of a set of n consecutive positive integers is m1. One of the  [#permalink]

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New post 06 Dec 2019, 03:05
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The mean of a data set of n consecutive positive integers is \(m_1\). One of the integers is removed from the set to create a new set of positive integers with a mean of \(m_2\). Is n even?

(1) \(m_1=m_2\)

(2) The least element of the original set of integers is odd.


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Re: The mean of a set of n consecutive positive integers is m1. One of the  [#permalink]

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New post 06 Dec 2019, 03:51
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(1) m1 = m2
e.g. N=5 with elements of the set {1,2,3,4,5}, so m1=3.
When N=4, m1=m2=3. This can only happen if N is odd and we remove the sole median of the set. Thus, n is surely NOT even.
SUFFICIENT

(2) The least element of the original set of the integers is odd.
We cannot determine whether n is even or odd, based on the information that the least element of the original set of the integers is odd.
NOT SUFFICIENT.

ANSWER IS (A)

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Re: The mean of a set of n consecutive positive integers is m1. One of the  [#permalink]

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New post 06 Dec 2019, 04:58
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Quote:
The mean of a data set of n consecutive positive integers is m_1. One of the integers is removed from the set to create a new set of positive integers with a mean of m_2. Is n even?

(1) m_1=m_2

(2) The least element of the original set of integers is odd.


The mean of a set of consecutive integers is an INTEGER when n is an ODD;
The mean of a set of consecutive integers is a NON-INTEGER when n is an EVEN.

(1) m_1=m_2 sufic

The new set consists of one element of the previous set, thus its mean is an integer;
So if m_2=integer=m_1 then n=odd.

(2) The least element of the original set of integers is odd. insufic

Ans (A)
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Re: The mean of a set of n consecutive positive integers is m1. One of the  [#permalink]

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New post 06 Dec 2019, 19:25
1
\(a_1+a_2+...+a_n= m_1*n\)
\(a_1+a_2+...+a_n= m_2*(n-1)\) (one of the integers removed from the set )

Is n even?

(Statement1): \(m_1=m_2\)
--> In order \(m_1=m_2\) to be true, the one integer removed from the set should be median and n should be always odd
(Always NO)--Sufficient

(2) The least element of the original set of integers is odd.
NO matter whether the least element of the set is odd. No info about what connection is there between \(m_1\) and \(m_2\)
insufficient

The answer is A.
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Re: The mean of a set of n consecutive positive integers is m1. One of the  [#permalink]

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New post 06 Dec 2019, 21:11
1
We are given that the mean of n consecutive numbers is m1. One of integers is removed to create a new set of positive integers. So, we now have n-1 integers in the set. We are to determine if n is even.

First of all, let's form two sets of consecutive integers: one set with n even and the other with n odd.
Let n=5, A: 1,2,3,4,5
Let n=6, B: 1,2,3,4,5,6
We know that m1 for set A=3 while that for B=3.5

As a general rule, the mean of any odd set of consecutive integers is always an integer, while that for an even set of consecutive integers is always a fraction or non-integer.

Statement 1: m1=m2
Sufficient.
Let's take Set A {1,2,3,4,5} and remove 3 from the set, we have a new set Aa {1,2,4,5}. The average of the new set is (1+2+4+5)/4 = 3. If you take out the mean of a set of numbers and that mean happens to be a member of the set of numbers, the rest of the numbers will still form another set with a mean equal to the mean of the original set. This is always true since the mean is that one number that can be used to replace all elements of the set.

For an odd set of consecutive integers, since the mean is always equal to the median of the set, taking out the media will create another set that has a mean equal to the original set. Hence, we can conclude based on statement 1 that n is odd and not even.

n cannot be even because any number out of n even consecutive numbers will result in a mean which is different from the mean of the original set.

Statement 2: The least element of the original set of integers is odd.
Insufficient.
Just knowing that the least element in a consecutive set of integers is not enough to determine if that set is odd or even. As can be seen above, both sets A and B have the least element as 1, which is odd. Meanwhile set A is composed of odd number of elements while set B is composed of even number of elements.

The answer is option A.
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The mean of a set of n consecutive positive integers is m1. One of the  [#permalink]

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New post Updated on: 09 Dec 2019, 00:51
#1
for set S= ( 1,2,3,4,5) and (1,2,4,5) we get m1=m2
only for odd set of integers this relation is possible , n is not even
sufficient

#2
The least element of the original set of integers is odd
no relation b/w m1 & m2 and we get m1 and m2 for n is odd and even
insufficient
IMO A
The mean of a data set of n consecutive positive integers is m1. One of the integers is removed from the set to create a new set of positive integers with a mean of m2. Is n even?

(1) m1=m2

(2) The least element of the original set of integers is odd.

Originally posted by Archit3110 on 07 Dec 2019, 02:27.
Last edited by Archit3110 on 09 Dec 2019, 00:51, edited 1 time in total.
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Re: The mean of a set of n consecutive positive integers is m1. One of the  [#permalink]

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New post 08 Dec 2019, 07:14
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>> Answer: A
I think the first statement alone is enough to arrive at the answer. If a number is removed from a set and the mean doesn't change. The removed number must equal mean (if it is less than mean, the new mean should be greater). So we know that mean of the original set is an integer.
If n is even, the mean of the original set should have .5 (for example the mean of 3, 4, 5, 6 is 3.5).
Therefore n is odd.
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Re: The mean of a set of n consecutive positive integers is m1. One of the   [#permalink] 08 Dec 2019, 07:14
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