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If there are more than two numbers in a certain list, is each of the

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New post Updated on: 27 Aug 2018, 05:49
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If there are more than two numbers in a certain list, is each of the numbers in the list equal to 0?

(1) The product of any two numbers in the list is equal to 0.
(2) The sum of any two numbers in the list is equal to 0.


Am unable to understand how the answer is B
from II u can have a positive and a negative number totaling to 0 eg. 1+(-1).
However if we combine both statements the value of all elements is 0.. Pl Help

Originally posted by devinawilliam83 on 25 Feb 2012, 02:23.
Last edited by Bunuel on 27 Aug 2018, 05:49, edited 2 times in total.
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New post 25 Feb 2012, 02:30
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devinawilliam83 wrote:
if there are more than 2 numbers in a list. is each of the number in the list equal to 0?
I. The product of any 2 numbers in the list is equal to 0
II. The sum of any 2 numbers in the list is 0

Am unable to understand how the answer is B
from II u can have a positive and a negative number totaling to 0 eg. 1+(-1).
However if we combine both statements the value of all elements is 0.. Pl Help


If there are more than two numbers in a certain list, is each of the numbers in the list equal to 0?

Note that we are told that there are more than 2 numbers in the list.

(1) The product of any two numbers in the list is equal to 0 --> it's certainly possible all numbers to equal to 0 but it's also possible one number to be different from 0 and all other numbers to equal to 0 (in this case the product of ANY two numbers in the list will also be equal to 0). Not sufficient.

(2) The sum of any two numbers in the list is equal to 0 --> as there are more than 2 numbers in the list then all numbers must equal to 0 (if we were not told that there are more than 2 numbers in the list then it would be possible to have a list like {-1, 1} but as there are more than 2 numbers then in order the sum of ANY two numbers in the list to be equal to 0 all numbers must equal to zero). Sufficient.

Answer: B.

P.S. Please DO NOT reword or shorten the questions you post.
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Re: If there are more than two numbers in a certain list, is each of the  [#permalink]

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New post 13 Mar 2012, 02:17
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devinawilliam83 wrote:
If there are more than two numbers in a certain list, is each of the numbers in the list equal to 0?
(1) The product of any two numbers in the list is equal to 0.
(2) The sum of any two numbers in the list is equal to 0.


When I saw the phrase, "more than two numbers in a list", I immediately thought I am going to pick 3-number set. Not more. Reading on, "does each number equal to 0"? So the 3-number set I picked was {0,0,1}.

Option 1 says, "the product of any two numbers equal to 0". In my list, it does. But right away I figured that, because the other two numbers are 0, the product of any two numbers will always be 0. INSUFFICIENT.

Option 2 says, "the sum of any two numbers equal to 0". Aha! With this condition, the 0's in the set will not force the result to be 0. So, only B. SUFFICIENT.
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New post 12 Jan 2013, 03:39
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kiyo0610 wrote:
If there are more than two numbers in a certain list, is each of the numbers in the list equal to 0?

(1)The product of any two numbers in the list is equal to 0
(2)The sum of any two numbers in the list is equal to 0


FROM ONE

we cant tell whether all set members are 0 or not because if the set contains odd number of elements then for 1 to be true then all elements must be zero but if the number of elements is even , we can ve one element as an intiger for example that is larger or less than 0 and the statment still will hold true ... insuff

from 2
this could only hold true if all elemnts are 0's
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Re: If there are more than two numbers in a certain list, is eac  [#permalink]

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New post 28 Apr 2015, 20:26
Hi All,

When dealing with questions that talk about groups of unknown numbers, it often helps to come up with some examples that fit the limited information that you have. In that way, you can TEST VALUES by considering what the group of numbers COULD contain.

Here, we're told that the group of numbers consists of MORE than 2 numbers (so 3 or more numbers). We're asked if EACH of the numbers is 0. This is a YES/NO question.

Fact 1: The product of any two numbers in the list is equal to zero

Since the product of ANY number and 0 is 0, this means that the list COULD contain a non-0 number....In the second option, choosing any 2 numbers WILL result in a product that = 0.

IF the group of numbers is....
{0, 0, 0} then the answer to the question is YES
{0, 0, 1] then the answer to the question is NO
Fact 1 is INSUFFICIENT

Fact 2: The sum of any two numbers in the list equal to 0.

Since we're dealing with MORE than 2 numbers, this Fact provides a specific 'restriction' - we CAN'T have ANY non-0 numbers because then would could end up with a sum that is NOT 0.

IF...
{0, 0, 0} then the answer to the question is YES

IF....
{-1, 1, 1} then we could end up with (1) + (1) = 2, which does NOT fit the given Fact. Thus, this example is NOT possible and neither is any other example that could lead to a Non-0 sum. By extension, that means that EVERY number in the group MUST be 0.
Fact 2 is SUFFICIENT

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Re: If there are more than two numbers in a certain list, is each of the  [#permalink]

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New post 19 Nov 2016, 18:24
What if there are four numbers. -1,-3 ,+1,+3. SUM is zero but each number is not.
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Re: If there are more than two numbers in a certain list, is each of the  [#permalink]

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New post 29 Mar 2017, 07:50
Hey everyone,
I have a more general question regarding this type of question.
So the question basically says "Is each of the numbers in the list equal to 0?"

Maybe I complicate myself, but I thought that statement I is sufficient, as it provides a clear answer to the question.
--> No, the numbers in the list are not equal to zero.

So my question is, if this kind of question pops up in the GMAT, does the sufficiency of the statements only depend on a positive affirmation of the question?

I hope I could express what I mean (since I'm not native in English :-))?!

Thanks,
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New post 29 Mar 2017, 08:00
Vincent89 wrote:
Hey everyone,
I have a more general question regarding this type of question.
So the question basically says "Is each of the numbers in the list equal to 0?"

Maybe I complicate myself, but I thought that statement I is sufficient, as it provides a clear answer to the question.
--> No, the numbers in the list are not equal to zero.

So my question is, if this kind of question pops up in the GMAT, does the sufficiency of the statements only depend on a positive affirmation of the question?

I hope I could express what I mean (since I'm not native in English :-))?!

Thanks,
Vincent


For (1):
It's certainly possible all numbers to equal to 0: for example {0, 0, 0} --> answer YES.

It's also possible one number to be different from 0 and all other numbers to equal to 0 (in this case the product of ANY two numbers in the list will also be equal to 0). For example, {0, 0, 1} --> answer NO.

As for your other question: in YES/NO DS questions a definite NO answer to the question is still considered to be sufficient.

Hope it's clear.
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Re: If there are more than two numbers in a certain list, is each of the  [#permalink]

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New post 19 Oct 2018, 08:14
devinawilliam83 wrote:
If there are more than two numbers in a certain list, is each of the numbers in the list equal to 0?

(1) The product of any two numbers in the list is equal to 0.
(2) The sum of any two numbers in the list is equal to 0.

\(L = \left\{ {\,{x_1}\,,\,{x_2}\,,\, \ldots \,\,,\,\,{x_n}} \right\}\,\,\,\,,\,\,\,n \geqslant 3\)

\(?\,\,\,:\,\,\,{\text{all}}\,\,{\text{zero}}\)

\(\left( 1 \right)\,\,\,{x_j} \cdot {x_k} = 0\,\,\,\,\,\left( {j \ne k} \right)\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,L = \left\{ {0,0, \ldots ,0,0} \right\}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,L = \left\{ {0,0, \ldots ,0,1} \right\}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\, \hfill \\
\end{gathered} \right.\)

What about statement (2)? Do you "feel" this statement is sufficient... but you cannot be 100% sure?

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\(\left( 2 \right)\,\,\left\{ \begin{gathered}
\,{x_j} + {x_k} = 0 \hfill \\
{x_k} + {x_m} = 0 \hfill \\
\end{gathered} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,\,\,{x_j} - {x_m} = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{x_j} = {x_m}\,\,\,\,{\text{for}}\,\,\,\underline {{\text{ANY}}} \,\,\,\,{x_j}\,,\,\,{x_k}\,,\,\,{x_m}\,\,\,{\text{in}}\,\,L\)

\(\,\left\{ \begin{gathered}
\,{x_j} = {x_m} \hfill \\
\,0 = {x_j} + {x_m} = 2\,\, \cdot {x_j} \hfill \\
\end{gathered} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{x_j} = 0\,\,\,{\text{for}}\,\,{\text{all}}\,\,j\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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New post 21 Oct 2018, 08:12
fskilnik wrote:
(2) The sum of any two numbers in the list is equal to 0.

\(\left( 2 \right)\,\,\left\{ \begin{gathered}
\,{x_j} + {x_k} = 0 \hfill \\
{x_k} + {x_m} = 0 \hfill \\
\end{gathered} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,\,\,{x_j} - {x_m} = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{x_j} = {x_m}\,\,\,\,{\text{for}}\,\,\,\underline {{\text{ANY}}} \,\,\,\,{x_j}\,,\,\,{x_k}\,,\,\,{x_m}\,\,\,{\text{in}}\,\,L\)

\(\,\left\{ \begin{gathered}
\,{x_j} = {x_m} \hfill \\
\,0 = {x_j} + {x_m} = 2\,\, \cdot {x_j} \hfill \\
\end{gathered} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{x_j} = 0\,\,\,{\text{for}}\,\,{\text{all}}\,\,j\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle\)

I was asked if there is another formal proof of the sufficiency of the statement (2) but in a more "down-to-earth" arguments.

Certainly!

Let´s imagine (at first) that there is a negative number among the elements in the given list, say A.
In this case there is another number in the list (say B) such that A+B= 0, hence B must be positive (B=-A).
Let´s consider any third number (say C) of the list. (We know the list has at least three elements.)
It is impossible to have A+C = 0 (C would be positive) and B+C = 0 (C would be negative) simultaneously,
therefore there is NO negative number among the elements of the given list.

Let´s now imagine that there is a positive number among the elements in the given list, say B.
In this case, there is a negative number (say A) so that B+A = 0 (A=-B), but we have already proven
(in the previous paragraph) that there are NO negative elements in the given list.

From both paragraphs above, we are sure all numbers (elements) in the given list must be non-negative
and also non-positive, hence all of them are equal to zero.

Regards,
Fabio.
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Re: If there are more than two numbers in a certain list, is each of the  [#permalink]

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New post 14 May 2019, 21:36
rahuljain01408@gmail.com wrote:
What if there are four numbers. -1,-3 ,+1,+3. SUM is zero but each number is not.


True, Sum of any two numbers is zero
-1,1
3,-3
But your assumption fails when we take -1 and 3, their sum is not zero. So sum of any two numbers zero is only possible iff the list contains all zeroes.
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Re: If there are more than two numbers in a certain list, is each of the  [#permalink]

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New post 14 Sep 2019, 22:30
thanks a lot everyone,
I'm one person who is confused why the answer is not "A" but "B".
I think most students who found it a problem because we failed to focus the question in the part "each of the numbers in the list is 0...."
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Re: If there are more than two numbers in a certain list, is each of the   [#permalink] 14 Sep 2019, 22:30
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