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

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

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

Final Answer:

GMAT assassins aren't born, they're made,
Rich

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.

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.

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

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

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

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.

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.

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.

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...."

Target question: Is each of the numbers n the list equal to zero?

Given: There are more than 2 numbers in the list

Statement 1: The product of any 2 numbers in the list is ZERO
There are several possible sets that satisfy this condition. Here are two:
Case a: the set is {0, 0, 0} in which case every number in the list is equal to ZERO
Case b: the set is {0, 0, 1} in which case every number in the list is not equal to ZERO
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The sum of any 2 numbers in the list is ZERO
This statement ensures that every number in the list is equal to ZERO
Here's why:
Let's say the number k is in the set.
If any two numbers add to zero, then -k must be another number in the set.
At this point, we could have a set like {1, -1} where the numbers do not equal zero. Or we could have a set like {0, 0} where the numbers do equal zero. HOWEVER, we are told that the set has more than 2 numbers.
So, what does a third value look like?
Well, if we already have k in the set, then the third value must also be -k, otherwise we wouldn't get a sum of zero if we picked k and the third value.
At this point, we know that that the set must contain: k, -k, -k [and possibly more values]
Now let's examine the pair of values -k and -k
If we add them, we get -2k. The ONLY way that this sum can equal zero if for k to equal zero.
We can extend this logic to conclude that EVERY value in the set must equal ZERO
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent

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.
x = 0, y = 0, z = 0 YES
x = 0, y = 0, z = 1 NO
Insufficient

(2) The sum of any two numbers in the list is equal to 0.
Every term must be equal to zero since ANY two numbers in the list sums to 0.
Sufficient.

B.

Video solution from Quant Reasoning starts at 18:20
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why can't
{1, -1, 1, -1, 1} suits option B's description?

the first term +second = 0
second term + third term = 0
... ...
fourth term + fifth term = 0

Help... I'm going crazy

(2) says "The sum of ANY two numbers in the list is equal to 0". This means that the sum of ALL possible pairs from the list is 0. Your list is not valid becasue if you pick 1 and 1 OR -1 and -1, then their sums are not 0. As discussed above (for example HERE), for the sum of ANY two numbers in the list to be equal to 0, all numbers must be 0.

Hope it helps.

I just got this question on mock test, got tricked as well, here is my logic to come to B

Supppose we have 3 numbers: A B C

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

It can be All ABC = 0 or
it just happen 1 of the number (A) = 0 --> AB & AC = 0
But wait BC is not 0 yet, so at least B also 0, then C does not have to be 0 to fulfill
INSUFFICIENT

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

It can be All ABC = 0 or
A= 1 & B = -1 = A+B = 0
But wait what if B + C = 0 then C = 1 as well
This leaves A + C = 2
So in order sum of any two numbers = 0 they all gotta be 0