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# A certain list consist of several different integers. Is the product

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Re: A certain list consist of several different integers. Is the product [#permalink]
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Dear Bunuel:

I am sorry still i am not clear.

Can you explain how are you sure From ST 1 that all are either - or +? We just know smallest and largest number's multiplication is positive. if we combine ST 1 & 2, it may be

(i.e, lets say 6 even numbers)

- - - - + - = -
+ + - - + + = +
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Re: A certain list consist of several different integers. Is the product [#permalink]
Quote:
(1)+(2) Now if we have scenario A (from 1) then answer is YES. If we have scenario B, then as there are even number of integers (from 2) the product of all integers still would be positive, so answer is still YES. Sufficient.

Hi, I have one question here. Statement 2 says, there are even number of integers in the list. How can we assume all are negative or positive? What if, (-,-,-,+). This will result in negative.

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Re: A certain list consist of several different integers. Is the product [#permalink]
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pgmat
Quote:
(1)+(2) Now if we have scenario A (from 1) then answer is YES. If we have scenario B, then as there are even number of integers (from 2) the product of all integers still would be positive, so answer is still YES. Sufficient.

Hi, I have one question here. Statement 2 says, there are even number of integers in the list. How can we assume all are negative or positive? What if, (-,-,-,+). This will result in negative.

Answer to the question is C, not E.

(1)+(2): From statement (1) we have that either all integers are negative or all integers are positive (check this: a-certain-list-consists-of-several-different-integers-126040.html#p878206). Statement (2) says that there are even number of elements in the set. So in either of cases the product will be positive.

Hope it's clear.
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Re: A certain list consist of several different integers. Is the product [#permalink]
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Kudos
Statement 2 is insufficient, because it says there are even number of items.

But from the first statement the numbers are either on the negative side of the number line or positive side of the number line.
only then multiplying the larger and smaller will lead to a positive number.

Ignoring the positive side, because odd number of items or even number of items will lead to a positive outcome.

But for the negative side of the number line to become postive there should be an even number of multiples.
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Re: A certain list consist of several different integers. Is the product [#permalink]
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Kudos
To have a positive product, we must have an even number of negatives (0, 2, 4...) so that the negatives will cancel out in the multiplication.

REPHRASE: Are there an even number of negatives?

1) Max * Min is positive means Max and Min have the same sign. If they're both positive, then everything is positive and so is the product of all integers. However, if Max and Min are both negative, the product could be negative if we do NOT have an even number of negatives. Example {-3, -2, -1}. NOT SUFFICIENT.

2) By itself, this doesn't tell us whether there is an even number of negatives. Doesn't answer our rephrase.

Merge statements: (2) tells us that we have an even number of values. Since all the values have the same sign (1 says Max and Min have the same sign), either we have all positives or we have an even number of negatives. Either way, the product of all terms will be positive.

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Re: A certain list consist of several different integers. Is the product [#permalink]
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zz0vlb
A certain list consist of several different integers. Is the product of all integers in the list positive?

(1) The product of the greatest and smallest of the integers in the list is positive.
(2) There is an even number of integers in the list.

Here is my reasoning of the question:

Statement 1. (1) The product of the greatest and smallest of the integers in the list is positive.

take numbers 1, -2, 3

now as per 1st --> 1 *3 = 3 (positive) but the product of all integers is -6. hence not sufficient.

Statement 2. (2) There is an even number of integers in the list.

take numbers -1, -2, -3, 4 (so we have even # of integers)

The product of these numbers can be negative (-1) *(-2)* (-3)* (4) = - 24 or can be positive (1) *(-2)* (-3)* (4 ) =24

As you see from second statement for the product of integers to be positive the product of the greatest and smallest of the integers must be positive (in other words (the smallest and greatest integers must be positive)

Hence combining two options together is sufficient.

Option C YAY!
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Re: A certain list consist of several different integers. Is the product [#permalink]
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dave13
zz0vlb
A certain list consist of several different integers. Is the product of all integers in the list positive?

(1) The product of the greatest and smallest of the integers in the list is positive.
(2) There is an even number of integers in the list.

Here is my reasoning of the question:

Statement 1. (1) The product of the greatest and smallest of the integers in the list is positive.

take numbers 1, -2, 3

now as per 1st --> 1 *3 = 3 (positive) but the product of all integers is -6. hence not sufficient.

Statement 2. (2) There is an even number of integers in the list.

take numbers -1, -2, -3, 4 (so we have even # of integers)

The product of these numbers can be negative (-1) *(-2)* (-3)* (4) = - 24 or can be positive (1) *(-2)* (-3)* (4 ) =24

As you see from second statement for the product of integers to be positive the product of the greatest and smallest of the integers must be positive (in other words (the smallest and greatest integers must be positive)

Hence combining two options together is sufficient.

Option C YAY!

Unfortunately the reasoning above is not correct.

For (1): if the set is {1, -2, 3 }, then the smallest term there is -2 and the largest term is 3 --> the product = -6. So, this set is not possible. Also, when testing a statement, you should get both an YES and a NO answer to get insufficiency.

For (2): the product of -1, -2, -3, and 4 is -24 only.

For (1)+(2): we don't get that all the elements in the list must be positive. The elements in the list are either positive or negative but since the number of elements is even, then even if all the elements are negative, the product is still positive. So, in any case the product is positive.

I suggests to study the solutions above carefully. For example, this one: https://gmatclub.com/forum/a-certain-li ... ml#p729037
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Re: A certain list consist of several different integers. Is the product [#permalink]
Bunuel
A certain list consist of several different integers. Is the product of all integers in the list positive?

(1) The product of the greatest and smallest of the integers in the list is positive.
Two cases:
A. all integers in the list are positive: in this case product of all integers would be positive;
OR
B. all integers in the list are negative: now, if there is even number of integers, then product of all integers would be positive BUT if there is odd number of integers, then product of all integers would be negative.

Not sufficient.

(2) There is an even number of integers in the list.

(1)+(2) Now if we have scenario A (from 1) then answer is YES. If we have scenario B, then as there are even number of integers (from 2) the product of all integers still would be positive, so answer is still YES. Sufficient.

Hope it's clear.

Can't the integers be divided between +ve's and -ve's? Such as, total 8 integers for statement (1), 3 are negative and 5 are positive, which will give us negative product ?
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Re: A certain list consist of several different integers. Is the product [#permalink]
Bunuel
A certain list consist of several different integers. Is the product of all integers in the list positive?

(1) The product of the greatest and smallest of the integers in the list is positive.
Two cases:
A. all integers in the list are positive: in this case product of all integers would be positive;
OR
B. all integers in the list are negative: now, if there is even number of integers, then product of all integers would be positive BUT if there is odd number of integers, then product of all integers would be negative.

Not sufficient.

(2) There is an even number of integers in the list.

(1)+(2) Now if we have scenario A (from 1) then answer is YES. If we have scenario B, then as there are even number of integers (from 2) the product of all integers still would be positive, so answer is still YES. Sufficient.

Hope it's clear.

WHAT IF THE LIST HAS 4 NUMBERS LIKE ---> (-),(+),(-),(-)?
THE RESULT IS NEGATIVE???
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Re: A certain list consist of several different integers. Is the product [#permalink]
taniachauhan
Bunuel
A certain list consist of several different integers. Is the product of all integers in the list positive?

(1) The product of the greatest and smallest of the integers in the list is positive.
Two cases:
A. all integers in the list are positive: in this case product of all integers would be positive;
OR
B. all integers in the list are negative: now, if there is even number of integers, then product of all integers would be positive BUT if there is odd number of integers, then product of all integers would be negative.

Not sufficient.

(2) There is an even number of integers in the list.

(1)+(2) Now if we have scenario A (from 1) then answer is YES. If we have scenario B, then as there are even number of integers (from 2) the product of all integers still would be positive, so answer is still YES. Sufficient.

Hope it's clear.

WHAT IF THE LIST HAS 4 NUMBERS LIKE ---> (-),(+),(-),(-)?
THE RESULT IS NEGATIVE???

From statement (1), we got to know that either, the set has all even even positive numbers or negative numbers.
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Re: A certain list consist of several different integers. Is the product [#permalink]
lakshya14
taniachauhan
Bunuel
A certain list consist of several different integers. Is the product of all integers in the list positive?

(1) The product of the greatest and smallest of the integers in the list is positive.
Two cases:
A. all integers in the list are positive: in this case product of all integers would be positive;
OR
B. all integers in the list are negative: now, if there is even number of integers, then product of all integers would be positive BUT if there is odd number of integers, then product of all integers would be negative.

Not sufficient.

(2) There is an even number of integers in the list.

(1)+(2) Now if we have scenario A (from 1) then answer is YES. If we have scenario B, then as there are even number of integers (from 2) the product of all integers still would be positive, so answer is still YES. Sufficient.

Hope it's clear.

WHAT IF THE LIST HAS 4 NUMBERS LIKE ---> (-),(+),(-),(-)?
THE RESULT IS NEGATIVE???

From statement (1), we got to know that either, the set has all even even positive numbers or negative numbers.

I AM SO SORRY IM STILL NOT GETTING IT.
SO AS PER STATEMENT 1, THE PRODUCT OF SMALLEST AND LARGEST DIGIT SHOULD BE POSITIVE.
NOW ASSUME, WE HAVE A LIST OF n NUMBERS, WHICH ARE ARRANGED (JUST FOR UNDERSTANDING)
NOW, IF n=2, BOTH NUMBERS ARE EITHER NEGATIVE OR POSITIVE TO MAKE SURE OUR STATEMENT 1 IS CORRECT AS GIVEN .
IF n=3 or n=5, THE FIRST AND LAST DIGIT IS EITHER BOTH -VE OR BOTH +VE. BUT WHAT IF IN BETWEEN WE HAVE (+,-,+)
THUS, IT WILL BE LIKE (-SMALEST) *+*-*+* (-LARGEST) ??
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Re: A certain list consist of several different integers. Is the product [#permalink]
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taniachauhan
lakshya14
taniachauhan
Bunuel
A certain list consist of several different integers. Is the product of all integers in the list positive?

(1) The product of the greatest and smallest of the integers in the list is positive.
Two cases:
A. all integers in the list are positive: in this case product of all integers would be positive;
OR
B. all integers in the list are negative: now, if there is even number of integers, then product of all integers would be positive BUT if there is odd number of integers, then product of all integers would be negative.

Not sufficient.

(2) There is an even number of integers in the list.

(1)+(2) Now if we have scenario A (from 1) then answer is YES. If we have scenario B, then as there are even number of integers (from 2) the product of all integers still would be positive, so answer is still YES. Sufficient.

Hope it's clear.

WHAT IF THE LIST HAS 4 NUMBERS LIKE ---> (-),(+),(-),(-)?
THE RESULT IS NEGATIVE???

From statement (1), we got to know that either, the set has all even even positive numbers or negative numbers.

I AM SO SORRY IM STILL NOT GETTING IT.
SO AS PER STATEMENT 1, THE PRODUCT OF SMALLEST AND LARGEST DIGIT SHOULD BE POSITIVE.
NOW ASSUME, WE HAVE A LIST OF n NUMBERS, WHICH ARE ARRANGED (JUST FOR UNDERSTANDING)
NOW, IF n=2, BOTH NUMBERS ARE EITHER NEGATIVE OR POSITIVE TO MAKE SURE OUR STATEMENT 1 IS CORRECT AS GIVEN .
IF n=3 or n=5, THE FIRST AND LAST DIGIT IS EITHER BOTH -VE OR BOTH +VE. BUT WHAT IF IN BETWEEN WE HAVE (+,-,+)
THUS, IT WILL BE LIKE (-SMALEST) *+*-*+* (-LARGEST) ??

Statement (1) says, The product of the greatest and smallest of the integers in the list is positive. Which means there are "n" integers, but the product of the smallest and the reatest is positive.

Now, the product of 2 integers can only be positive iff both the integers are either positive or negative. If both of them are negative then the last integer in the set would be a negative integer and the list of set would be over in negative range. Same stands for positive integers. Because of statement (1), we cannot mix "positives" and "negatives".

M= {a, b, c, d, e, f, g} let the smallest be "a" and the greatest be "g".
Now with statement (1), a x g = positive, if "a" is positive then "g" has to be positive to make the product positive, and also since set M is in ascending order.

Now, if a x g = negative, and this time "a" is negative, then in order to make the product positive, we need our greatest integer "g" to be negative only. Also, notice "g" is our greatest number in the list , which means thats the end of the list and we can't go to the positive side of the number line, which would make the greatest number a positive with "a" still remaining negative, giving a negative product. This will violate the statement (1).

Hope that's clear.
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A certain list consist of several different integers. Is the product [#permalink]
Bunuel
A certain list consist of several different integers. Is the product of all integers in the list positive?

(1) The product of the greatest and smallest of the integers in the list is positive.
Two cases:
A. all integers in the list are positive: in this case product of all integers would be positive;
OR

B. all integers in the list are negative: now, if there is even number of integers, then product of all integers would be positive BUT if there is odd number of integers, then product of all integers would be negative.

Not sufficient.

(2) There is an even number of integers in the list.

(1)+(2) Now if we have scenario A (from 1) then answer is YES. If we have scenario B, then as there are even number of integers (from 2) the product of all integers still would be positive, so answer is still YES. Sufficient.

Hope it's clear.

If we take -1, -2, -3 and +4 then answer would be Negative. So the answer should be E actually. Because it gives an unstable answer after combining the statements.
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Re: A certain list consist of several different integers. Is the product [#permalink]
The trick here is to realize that S1 gives us the indirect implication that all numbers are -ve.

With that logic, all then we have to figure out is the # of elements in the set being even / odd, which S2 provides us.

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Re: A certain list consist of several different integers. Is the product [#permalink]
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