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

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

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Updated on: 27 Aug 2018, 04:49
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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, 01:23.
Last edited by Bunuel on 27 Aug 2018, 04:49, edited 2 times in total.
Edited the question
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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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25 Feb 2012, 01:30
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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.

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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13 Mar 2012, 01:17
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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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Re: If there are more than two numbers in a certain list, is each of the  [#permalink]

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12 Jan 2013, 02:39
1
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
B
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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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13 Jul 2013, 09:11
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.

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

B is the answer. If it is the condition that any of the two numbers have sum zero, then they all have to be zero only.
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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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21 Jul 2014, 03:24
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.

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

It is important to understand what the statements say here. They each say ANY 2 numbers, which means that no matter how many numbers you have and which of these numbers you take, you always get the result as stated in the statements. Let's take a look:

(1) The product of any 2 numbers = 0. If you have 100 numbers and 99 are 0 and 1 is not you will ahve any 2 numbers = 0. If you have 100 = 0 you get the same result. IS.
(2) The sum of any 2 numbers = 0. This can only mean that every number = 0. You are right that 1+ (-1) = 0 bit if you have a third number, e.g. 0 or 3 or 9. you will get a result different from 0. Hence the answer can only be that all the numbers are 0s. Hence B.
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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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19 Nov 2016, 17:24
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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20 Nov 2016, 02:16
rahuljain01408@gmail.com wrote:
What if there are four numbers. -1,-3 ,+1,+3. SUM is zero but each number is not.

The second statement says that "The sum of ANY two numbers in the list is equal to 0", which is not true for your list.
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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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29 Mar 2017, 06:50
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,
Vincent
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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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29 Mar 2017, 07:00
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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19 Oct 2018, 07:14
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?

EMBRACE MATHEMATICS and develop your quantitative maturity to EXCEL IN YOUR EXAM (and in the MBA that goes right after it)!

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

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

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

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

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19 Oct 2018, 22:44
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.

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

I find picking numbers as the best approach here.

statement 1:
So lets assume number as 1 ,0 ,0
Clearly the product of any numbers is zero here but not all numbers are zero here.

statement 2:

So lets assume number set again as as 1 ,0 ,0

The sum cant be zero if you do not have all zero's

For a set with equal weighted distribution on both sides across number line (from 0 origin)

-1,0,1 or -4,-2,0,2,4 -6,0,2,4 (here sum of any two numbers can be proved as not zero...say first set -1,0,1 (0+1)=not zero)

Hence every element has to be zero

So,B 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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21 Oct 2018, 07:12
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 &nbs [#permalink] 21 Oct 2018, 07:12
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