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# If list S contains nine distinct integers, at least one of w

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If list S contains nine distinct integers, at least one of w  [#permalink]

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24 Jun 2013, 02:07
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95% (hard)

Question Stats:

34% (02:06) correct 66% (02:17) wrong based on 487 sessions

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If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.

(2) The sum of all nine integers in list S is equal to the median of list S.

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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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24 Jun 2013, 08:43
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Rock750 wrote:
If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.

(2) The sum of all nine integers in list S is equal to the median of list S.

Let the set of 9 distinct integers in increasing order be$$a_1$$ , $$a_2$$ ... $$a_9$$.

From F.S 1, we know that $$a_1*a_2*....a_4*a_5....a_9$$ = $$a_5$$ $$\to$$ $$a_5(a_1*a_2*...a_4....a_9-1)$$ = 0.

Thus, either $$a_5$$ = 0 OR $$(a_1*a_2*...*a_5....a_9-1)$$ = 0, the latter is not possible as no product of 8 distinct integers can ever equal 1.

Thus, the median,$$a_5$$ = 0 and not positive. Sufficient.

From F. S 2, for -4,-3,-2,-1,0,1,2,3,4,the median is 0 and a NO for the question stem. Again, for the series -10,-3,-1,0,1,2,3,4,5, the median is 1, which is positive, and hence a YES for the question stem. Thus, Insufficient.

A.
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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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24 Jun 2013, 13:05
mau5 wrote:
Rock750 wrote:
If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.

(2) The sum of all nine integers in list S is equal to the median of list S.

Let the set of 9 distinct integers be$$a_1$$ , $$a_2$$ ... $$a_9$$.

From F.S 1, we know that $$a_1*a_2*....a_4*a_5....a_9$$ = $$a_5$$ $$\to$$ $$a_5(a_1*a_2*...a_5....a_9-1)$$ = 0.

Thus, either $$a_5$$ = 0 OR $$(a_1*a_2*...*a_4....a_9-1)$$ = 0, the latter is not possible as no product of 8 distinct integers can ever equal 1.

Thus, the median,$$a_5$$ = 0 and not positive. Sufficient.

From F. S 2, for -4,-3,-2,-1,0,1,2,3,4,the median is 0 and a NO for the question stem. Again, for the series -10,-3,-1,0,1,2,3,4,5, the median is 1, which is positive, and hence a YES for the question stem. Thus, Insufficient.

A.

How did u come up with this : $$a_1*a_2*....a_4*a_5....a_9$$ = $$a_5$$ $$\to$$ $$a_5(a_1*a_2*...a_4....a_9-1)$$ = 0. ?
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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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24 Jun 2013, 23:28
Rock750 wrote:

How did u come up with this : $$a_1*a_2*....a_4*a_5....a_9$$ = $$a_5$$ $$\to$$ $$a_5(a_1*a_2*...a_4....a_9-1)$$ = 0. ?

Could you elaborate on your doubt? The above equation is what is given in the Fact Statement.
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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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25 Jun 2013, 00:10
mau5 wrote:
Rock750 wrote:

How did u come up with this : $$a_1*a_2*....a_4*a_5....a_9$$ = $$a_5$$ $$\to$$ $$a_5(a_1*a_2*...a_5....a_9-1)$$ = 0. ?

Could you elaborate on your doubt? The above equation is what is given in the Fact Statement.

The first equation is given in the fact statement OK But what about the transition from the first to the second ?

I think u factorized by a_5 from both the two parts of the equation which must lead to the following :

$$a_1*a_2*....a_4*a_5....a_9$$ = $$a_5$$ $$\to$$ $$(a_1*a_2*...a_4....a_9)$$ = 1.

Am I missing something here ?
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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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25 Jun 2013, 00:59
Rock750 wrote:

The first equation is given in the fact statement OK But what about the transition from the first to the second ?

I think u factorized by a_4 from both the two parts of the equation which must lead to the following :

$$a_1*a_2*....a_4*a_5....a_9$$ = $$a_5$$ $$\to$$ $$(a_1*a_2*...a_5....a_9)$$ = 1.

Am I missing something here ?

Yes, we cannot cancel $$a_5$$ from both sides because $$a_5$$ can be zero.
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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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31 Oct 2013, 00:28
For statement 1)

Can we deduce that since at least 1 member of the set must be negative, that would give a no answer to the stem of the question.

Ie, since at least one member is negative, the only possible answers we could get are either are 0 or <0.

Which would give a no answer to the question.

Or is this reasoning incorrect for this exam.

Thanks,
hunter
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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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19 Apr 2017, 17:39
Rock750 wrote:
If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.

(2) The sum of all nine integers in list S is equal to the median of list S.

Statement 1:

Two possible numbers to start with are 1 and 0. We cannot have a set of 9 numbers where product of all numbers equals the median if we use 1 (e.x --3 x -2 x -1 x 1 x 2 x 3 x4 ); however, any number times 0 is 0 so if we use 0 as a median we can simply take any product of four negative numbers and multiply them by the product of four positive numbers and then by zero to yield a number equal to the mean or vice versa the product of any four negative numbers times zero times the product of any four positive numbers equals 0 a.k.a the median

Sufficient.

Statement 2

We could have a set of numbers where 0 is the median (1,2,3,4,0,-1,-2,-3,-4) the result of which would be the media- however, we could also have --6,-4,-3,-2,-1,2,3,5,7 which would equal -1 which is also the median of that set and a negative number

Insufficient
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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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03 Sep 2017, 01:42
Rock750 wrote:
If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.

(2) The sum of all nine integers in list S is equal to the median of list S.

Statement 1: Product if integers = Median
Which is true only if either all terms are 1 or ,-1 or Median is zero

Since integers are distinct so median has to be zero
Sufficient

Statement 2: set may be

(-5,-4,-3,-2,1,2,3,4,5) Or
(-5,-4,-3,-2,0,2,3,4,5)

Hence median may be 1 or zero or likewise
Not sufficient

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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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11 Sep 2017, 01:32
Rock750 wrote:
If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.

(2) The sum of all nine integers in list S is equal to the median of list S.

Tricky but you just to be mindful of the possibilities -9 integers- all must be different

St 1

The only way this scenario could hold true is if the median were 0

suff

St 2

You could actually have different scenarios as demonstrated below

-5,-4,-3,-2,1,2,3,4,5

-5,-4,-3,-2,-1,1,2,5,6

insuff

A
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If list S contains nine distinct integers, at least one of w  [#permalink]

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11 Feb 2018, 09:02
Statement 1:
Prod of all integers is equal median
Let prod = product of all integers except median.

given, $$prod$$ * $$median$$ = $$median$$
$$prod$$ * $$median$$ - $$median$$ = 0
$$median$$ * ($$prod$$ - 1) = 0 => median = 0 or prod = 1
but product of all integers except median cannot be 1, as all are distinct => median = 0 => not positive => sufficient

Statement 2:
Sum of all nine integers to equal to median.
case 1: we can have median as positive, sum of left 4 integers and right 4 integers to cancel each other
case 2: we can have median as negative, sum of left 4 integers and right 4 integers to cancel each other
case 3: we can have median as 0, sum of left 4 integer and right 4 integers to cancel each other.
insuff

(A)
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If list S contains nine distinct integers, at least one of w  [#permalink]

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06 Jun 2018, 21:23
-- EASY SOLUTION --

STATEMENT 1
x1 is the product of the 4 numbers left to the median
M is the median
x2 is the product of the 4 numbers right to the median

We have x1*x2*M=M or M(x1*x2-1)=0
(i) If M=0 it is a NO
(ii) If x1*x2=1 means that either of x1, x2 is 1 or either is -1. This is impossible since all the numbers must be different. So we cancel option (ii).

SUFFICIENT

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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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07 Jun 2018, 04:15
But if there are even no. of negatives numbers in the , the product of all 9 integers will be positive and the median will be positive
where as if there is an odd no of negative numbers in the set, the product will be negative and hence the mean will also be negative.
how is A sufficient?
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Re: If list S contains nine distinct integers, at least one of w  [#permalink]

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07 Jun 2018, 04:15
But if there are even no. of negatives numbers in the , the product of all 9 integers will be positive and the median will be positive
where as if there is an odd no of negative numbers in the set, the product will be negative and hence the mean will also be negative.
how is A sufficient?
Re: If list S contains nine distinct integers, at least one of w &nbs [#permalink] 07 Jun 2018, 04:15
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