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A certain list consists of several different integers
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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.
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Originally posted by zz0vlb on 28 Apr 2010, 07:28.
Last edited by Bunuel on 08 Aug 2012, 05:32, edited 1 time in total.
OA added.




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Re: Stuck with an easy number property....:(
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23 Feb 2011, 18:26
moniralinda wrote: 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)
    +  =  + +   + + = + (1) says: The product of the greatest and smallest of the integers in the list is positive. Product of two multiple to be positive they must have the same sign: So either: smallest * greatest = negative * negative and in this case as both the smallest and the greatest are negative then ALL integers in the list are negative OR smallest * greatest = positive * positive and in this case as both the smallest and the greatest are positive then ALL integers in the list are positive. Hope it's clear.
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Re: A certain list consists of several different integers
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02 Jul 2018, 03:23
dave13 wrote: zz0vlb wrote: 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/acertainli ... ml#p729037
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Re: Stuck with an easy number property....:(
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24 May 2010, 11:00
Ithink both statement together is not sufficient, so what is the real answer here? why do care about the existence of 0 ?



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Re: Stuck with an easy number property....:(
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24 May 2010, 11:43
diemxua175 wrote: Ithink both statement together is not sufficient, so what is the real answer here? why do care about the existence of 0 ? 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. Clearly insufficient. {2, 2}  answer NO; {2,4}  answer YES. (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. Answer: C. Hope it's clear.
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Re: Stuck with an easy number property....:(
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23 Feb 2011, 18:06
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: Stuck with an easy number property....:(
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23 Feb 2011, 20:01
Ohh.... Genius...Thanks a lot! Now I understood how come all have the same sign...Thank u sooo much.



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Re: Product of integers  Gmatprep
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04 Jan 2012, 02:09
Hi Janealams, 1. The above DS question is a Yes/No type Question. 2. Question Stem: Products of all integers in a list positive? => [a, b, c, r,t,d] or [1, 2, 9, 3]  a few random numbers in a set 3. Statement I: The product of Greatest and smallest integer in the List is +ve => Case 1: all integers in the list are +ve: [1, 2, 3, 4] => 4*1 = 4 = +ve  YES => Case 2: all integers in the list are ve: [1, 2, 3,4]  => (4)*(1) = 4 = +ve  YES => Case 3: few integers in the list are ve: [1, 2, 3,4]  => (4)*(1) = 4 = ve  NO Hence Statement I is Not Sufficient. 4. Statement 2: There is an even number of Integers in the list => We know from number properties that ve no multiplied even number of times is +ve and +ve no multiplied even number of times is +ve Ex: 2 * 2 * 3*7 = +ve and 2 * 7 * 9 *10 = +ve This statement provides only information about the Integers in the Set and this cant be helpful in determining the Question stem. So NO  In Sufficient. 5. Now Taking I and II together. => We know that from I  All numbers in set are either +ve or ve and from II that there are even number of Integers. Hence we can use the above information to get the value if the product of all Integers is +ve or not. Do let me know, if you need further explanation. Thanks, Arvind.
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Re: Product of integers  Gmatprep
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04 Jan 2012, 14:07
Stem rephrase: There are even number of negative numbers or all are positive. 1. Not sufficient information on the middle numbers, insufficient 2. Even numbers does not mean even negatives or all positives, insufficient Together, no overlapping information, so insufficient. E
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Re: A certain list consists of several different integers
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04 May 2012, 19:37
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. I think answer should be E. Could you please explain. Thanks.



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Re: A certain list consists of several different integers
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04 May 2012, 21:48
pgmat wrote: 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. I think answer should be E. Could you please explain. Thanks. 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: acertainlistconsistsofseveraldifferentintegers126040.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 consists of several different integers
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05 May 2012, 10:04
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: Product of integers  Gmatprep
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02 Jul 2013, 00:02
arvindravulavaru wrote: Hi Janealams,
1. The above DS question is a Yes/No type Question. 2. Question Stem: Products of all integers in a list positive? => [a, b, c, r,t,d] or [1, 2, 9, 3]  a few random numbers in a set 3. Statement I: The product of Greatest and smallest integer in the List is +ve => Case 1: all integers in the list are +ve: [1, 2, 3, 4] => 4*1 = 4 = +ve  YES => Case 2: all integers in the list are ve: [1, 2, 3,4]  => (4)*(1) = 4 = +ve  YES => Case 3: few integers in the list are ve: [1, 2, 3,4]  => (4)*(1) = 4 = ve  NO Hence Statement I is Not Sufficient.
Thanks, Arvind. Isn't this case incorrect as the stem says the product of the largest and smallest integer is positive? Is there an example where the list of largest and smallest is positive and the product of the integers can be negative?
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Re: Product of integers  Gmatprep
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02 Jul 2013, 00:11
fozzzy wrote: arvindravulavaru wrote: Hi Janealams,
1. The above DS question is a Yes/No type Question. 2. Question Stem: Products of all integers in a list positive? => [a, b, c, r,t,d] or [1, 2, 9, 3]  a few random numbers in a set 3. Statement I: The product of Greatest and smallest integer in the List is +ve => Case 1: all integers in the list are +ve: [1, 2, 3, 4] => 4*1 = 4 = +ve  YES => Case 2: all integers in the list are ve: [1, 2, 3,4]  => (4)*(1) = 4 = +ve  YES => Case 3: few integers in the list are ve: [1, 2, 3,4]  => (4)*(1) = 4 = ve  NO Hence Statement I is Not Sufficient.
Thanks, Arvind. Isn't this case incorrect as the stem says the product of the largest and smallest integer is positive? Is there an example where the list of largest and smallest is positive and the product of the integers can be negative?Yes, this solution is not clear. Proper solution is here: acertainlistconsistsofseveraldifferentintegers126040.html#p729037
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Re: A certain list consists of several different integers
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02 Jul 2013, 00:23
Thanks! 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. all positive 1,2,3,4 largest is 4, smallest is 1 product positive. When we multiply all the numbers the result is positive YES
all negative 1,2,3,4 largest is 1 smallest is 4 product is positive When we multiply all the numbers YESthird scenario 1,2,3 Largest 1, smallest 3 product is positive but in this case when we multiply all the numbers we get negative result so its NO (2) There is an even number of integers in the list. no constraint here it could be 1,2,3,4 in this case YES 1,2,3,4 product is negative so answer is NO Combined 1+2 we have the case highlighted in Blue so its sufficient! Answer C
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Re: A certain list consists of several different integers
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02 Jul 2013, 00:25
fozzzy wrote: Thanks!
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.
all positive 1,2,3,4 largest is 4, smallest is 1 product positive. When we multiply all the numbers the result is positive YES
all negative 1,2,3,4 largest is 1 smallest is 4 product is positive When we multiply all the numbers YES
third scenario 1,2,3 Largest 1, smallest 3 product is positive but in this case when we multiply all the numbers we get negative result so its NO
(2) There is an even number of integers in the list.
no constraint here it could be 1,2,3,4 in this case YES
1,2,3,4 product is negative so answer is NO
Combined 1+2 we have the case highlighted in Blue so its sufficient! Answer C Yes, that's correct.
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Re: Stuck with an easy number property....:(
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02 Jul 2013, 00:35
Bunuel wrote: moniralinda wrote: 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)
    +  =  + +   + + = + (1) says: The product of the greatest and smallest of the integers in the list is positive. Product of two multiple to be negative they must have the same sign: So either: smallest * greatest = negative * negative and in this case as both the smallest and the greatest are negative then ALL integers in the list are negative OR smallest * greatest = positive * positive and in this case as both the smallest and the greatest are positive then ALL integers in the list are positive. Hope it's clear. GENIUS. Newbie here but I'm already tired of giving you kudos. What a legend.



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Re: A certain list consists of several different integers
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17 Aug 2014, 15:44
zz0vlb wrote: 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. (1): the greatest and smallest integer could be both negative or both positive > insufficient (2): even number of integers does not point out the sign of the product > insufficient (1) + (2): all number are positive or negative and even number > both the products are positive > sufficient C is the answer
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A certain list consists of several different integers
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04 Aug 2016, 22:45
Hi All,
I always find it extremely useful whenever I am able to rephrase the question and simplify what is being asked.
A certain list contains several different integers. Is the product of the integers in the list positive?  Lets list conditions wherein the product will be positive ( also do let me know if I have missed any thing else)
1. All the number are positive 2. All number negative and the even number of terms ( neg * neg = positive )> 2 negatives make one positive. 3. Mixed set > Again even number of negatives. We dont really have to bother about the number of positive terms
P.s: 0 is not a positive integer, so also look out for any information that says one of the terms is 0. In which case the product will NOT be positive.
Now lets look at the statements:
1. The product of the greatest and the smallest of the integers in the list is positive
This means that both the terms are of the same sign:
a) Both are negatives, this means that all the terms are negative. In this case we need to number of terms to be even. Refer to point number 2 in the question analysis. b) Both are positives, All terms positive.
Since we dont know the exact sign this statement is insufficient.
(2) There is an even number of integers in the list
Now since we dont know which category the set belongs to ( of the 3 listed above), we cant be sure. For example:
a) 3 negatives , 5 positives ( total 8, even number of terms ) > product is negative b) 4 negatives, 4 positives ( total 8, even number of terms )>product is positive
Hence insufficient.
Now combining 1 & 2:
a) Both are negatives, this means that all the terms are negative & Even number of terms > product is even b) Both are positives, All terms positive & Even number of terms > product is even
Hence answer is C
Hope this helps!
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Re: A certain list consists of several different integers
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07 Feb 2017, 06:49
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. The answer is C
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