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From the consecutive integers -10 to 10 inclusive, 20 integers are ran

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Bunuel
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Re: From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   05 Jan 2018, 08:31
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dave13 wrote:
Bunuel wrote:
catty2004 wrote:
From the consecutive integers -10 to 10 inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers?

A. (-10)^20
B. (-10)^10
C. 0
D. –(10)^19
E. –(10)^20


Select 10 odd number of times and -10 the remaining number of times, for example select 10 once and -10 nineteen times, then the product will be \(10^1*(-10)^{19}=-10^{20}\).

Answer: E.

P.S. Please read and follow: https://gmatclub.com/forum/rules-for-pos ... 33935.html


Hello Bunuel, hope you have a fantastc day :)
Can you explain pleaseeee,what is the difference between (-10)^20 and –(10)^20 ? yes I see minus sign – that’s why I don’t understand…. :?
Also how can this number be the least number/ smallest one :? –(10)^20 if we multiply 10 TWENTY times by itself we get huge number... :?
Thanks!


\(-2^2 = -(2*2) = -4\).
\((-2)^2 = (-2)*(-2) = 4\).

\(-10^{20}\) is -(huge number), so it's has very small value. The same way as say, -100 is less than -10.
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Re: From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   16 Nov 2018, 00:56
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To get the smallest possible product, we need to consider the smallest number from this set: that would be \(-10\).

We are allowed to have duplicate numbers, but we have to be careful about choosing this number 20 times.

If we did, then we will get:
\((-10)^{20}\)


This ends up being a large, positive number. In order to get the smallest overall product, we need to think in terms of a negative solution.

By multiplying a large product by a negative number, we can find that negative product that we're looking for. There are two possibilities:

\(10 \times (-10)^{19}\\\\
10^{19} \times (-10)\)


The smallest possible product is \(–(10^{20})\).

The final answer is
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E
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Re: From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   18 Dec 2018, 12:16
catty2004 wrote:
From the consecutive integers -10 to 10 inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers?

A. (-10)^20
B. (-10)^10
C. 0
D. –(10)^19
E. –(10)^20


Each of these randomly selected integers could be a negative number, a positive number, or zero, so their product could be negative. Their product has the least possible value if it is a negative number with the greatest possible absolute value.

First, we select the number with the greatest possible absolute value for each place, except one. Then, we must be careful to select the right number for the last place because we have to consider not only its absolute value but also its sign.

Although -10 and 10 have the same absolute value, we should choose 10s for the first 19 places because negative numbers always complicate things. Since \(10^{19}\) is positive, we must select the negative number with the greatest possible absolute value for the last place in the product.

\(10^{19}(−10)=(−)10^{19}(10)=(−)10^{20}=−10^{20}\)

\(−10^{20}\) is the same number as \(−(10)^{20}\) in the correct answer choice.

Answer: E

If the given set of integers consisted of only non-negative or only non-positive numbers, then we would need different strategies.
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Re: From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   13 Dec 2020, 05:03
Bunuel, VeritasKarishma - what would be the answer to this question if repetition was not allowed? Would it be 0?
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Re: From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   14 Dec 2020, 22:26
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davidbeckham wrote:
Bunuel, VeritasKarishma - what would be the answer to this question if repetition was not allowed? Would it be 0?


Yes, you have 21 integers - 10 positive, 10 negative and 0.

You need to leave out 1 of these integers and find the product of other 20 in that case. If you leave out a positive integer, product is 0.
If you leave out a negative integer, product is 0.
If you leave out 0, product is a large positive integer because we have an even number of negative integers.
So least product in that case will be 0.
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Re: From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   27 May 2021, 01:32
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Concept: The question deals with Max/Min Concept.


Solution:


The integers are -10,-9,-8,-7,-6,-5,-4,-3,-2-1
0
1, 2 ,3 ,4 ,5 ,6 ,7 ,8 ,9 ,10

20 integers are randomly chosen with repetitions allowed=>We can choose any term and multiply it with the constraint that the maximum number of times we can choose it is 20.

=>To minimize the product, we have to maximize the value with "-" sign.

If we use (-10)*(-10)...............(20 times), we shall have a positive product and hence the idea of minimizing the product shall not be fulfilled.

If we use (-10)*(-10)..............(19 times), we shall have -10^19 as the largest negative product with 19 terms.
To maximize a negative term, multiply the largest positive value available and hence our 20th term can be (+10)

=>The minimum product shall be (-10)^19 * 10^1 = - { 10^1 * 10^19}

= -(10)^20 (option e)

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From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   Updated on: 15 Feb 2022, 06:12
KarishmaB wrote:
catty2004 wrote:
But i'm still not sure, why the other answer choices are eliminated and why E is the least possible answer?? >_<


Actually, the question means "smallest possible value of the product"

\(-10^{20}\) is the smallest value of the given 5 options. Since it is possible, so it is the answer. I guess you were thinking 'least possible' (as was I). The wording is definitely misleading.


What is the difference between "smallest possible" and "least possible" value of the product. Please explain KarishmaB Bunuel

Originally posted by lostminer on 12 Feb 2022, 07:52.
Last edited by lostminer on 15 Feb 2022, 06:12, edited 2 times in total.
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Re: From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   15 Feb 2022, 05:10
In order to determine the least possible product :
We need to select an odd number of negative numbers and the remaining numbers must be positive. So that the net product is negative in order to make sure the final product is a negative number.
In the negative numbers, the product is the least possible when the absolute value of the product is maximum.
Hence we need to maximise the net product. This can be done by choosing

-10, x number of times.
10, 20-x number of times such that the product is maximum possible and the sign of product is negative.
Hence the product will remain the same irrespective of the value of x.
The product is equal to \(\left(-10\right)^x\cdot\left(10\right)^{20-x}\ =\ -10^{20}\)



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From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   16 Feb 2022, 00:43
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AbhiR3 wrote:
KarishmaB wrote:
catty2004 wrote:
But i'm still not sure, why the other answer choices are eliminated and why E is the least possible answer?? >_<


Actually, the question means "smallest possible value of the product"

\(-10^{20}\) is the smallest value of the given 5 options. Since it is possible, so it is the answer. I guess you were thinking 'least possible' (as was I). The wording is definitely misleading.


What is the difference between "smallest possible" and "least possible" value of the product. Please explain KarishmaB Bunuel


AbhiR3

There isn't any difference here but there is a possibility of ambiguity in "least possible".

Least possible value could mean "smallest possible value" (least value that is possible) or least possible value could mean "the value that is not possible". The possibility of it being the value is least (or 0). "Least" is modifying "possible" in that case, not the value.
The way GMAT questions ask for "the best possible" answer even if only one option is the possible answer or "least likely the reason" for a statement that cannot be the reason.
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Re: From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   08 Jul 2023, 10:28
dave13 wrote:
Bunuel wrote:
catty2004 wrote:
From the consecutive integers -10 to 10 inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers?

A. (-10)^20
B. (-10)^10
C. 0
D. –(10)^19
E. –(10)^20


Select 10 odd number of times and -10 the remaining number of times, for example select 10 once and -10 nineteen times, then the product will be \(10^1*(-10)^{19}=-10^{20}\).

Answer: E.

P.S. Please read and follow: https://gmatclub.com/forum/rules-for-po ... 33935.html


Hello Bunuel, hope you have a fantastc day :)
Can you explain pleaseeee,what is the difference between (-10)^20 and –(10)^20 ? yes I see minus sign – that’s why I don’t understand…. :?
Also how can this number be the least number/ smallest one :? –(10)^20 if we multiply 10 TWENTY times by itself we get huge number... :?
Thanks!


(-10)^20 = +ve, see -10 has an even power. For example (-10)^3 = -1000, but (-10)^2 = +100. The least possible value of the product will be (-10)^19 multiplied by the highest positive integer in this series. That is 10. Therefore, (-the result will still be negative and the abs value of the product will be the highest, thus making it the lowest possible value of product. (-10)^19 * 10^1 = - (10)^20
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Re: From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   28 Aug 2023, 08:31
Bunuel wrote:
VeritasPrepKarishma wrote:
catty2004 wrote:
From the consecutive integers -10 to 10 inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers?

A. (-10)^20
B. (-10)^10
C. 0
D. –(10)^19
E. –(10)^20



The consecutive integers from -10 to 10 are: -10, -9, -8, -7 ... -1, 0, 1, ... 9, 10

You have to select 20 integers with repetition.

So you can select all 10. The product will be \(10^{20}\).
or
You can select all -10. The product will be \((-10)^{20}\) (which is essentially same as \(10^{20}\))
or
You can select ten 1s and ten -10s. The product will be \((-10)^{10}\)
or
You can select 0 and any other 19 numbers. The product will be 0.
or
You can select -1 and nineteen 10s. The product will be \(-(10)^{19}\).

But how will you get \(- (10)^{20}\)?
You need to select -1 and twenty 10s but you cannot select 21 numbers. You cannot have the product as negative \(10^{20}\).
Hence (E) is not possible.


E is possible if you select 10 odd number of times and -10 the remaining number of times. For example:

\(10^1*(-10)^{19}=-10^{20}\) or \(10^3*(-10)^{17}=-10^{20}\) or \(10^5*(-10)^{15}=-10^{20}\)... or \(10^{19}*(-10)^{1}=-10^{20}\).

So, the answer is E.



Hi Bunuel : can you please help me understand how -10^20 is negative integer & -2^2 is a positive integer , although they have the negative signs and even exponents. I’m little confused here.

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Bunuel
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From the consecutive integers -10 to 10 inclusive, 20 integers are ran [#permalink] New post   28 Aug 2023, 08:57
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MasteringGMAT wrote:
Bunuel wrote:
VeritasPrepKarishma wrote:

The consecutive integers from -10 to 10 are: -10, -9, -8, -7 ... -1, 0, 1, ... 9, 10

You have to select 20 integers with repetition.

So you can select all 10. The product will be \(10^{20}\).
or
You can select all -10. The product will be \((-10)^{20}\) (which is essentially same as \(10^{20}\))
or
You can select ten 1s and ten -10s. The product will be \((-10)^{10}\)
or
You can select 0 and any other 19 numbers. The product will be 0.
or
You can select -1 and nineteen 10s. The product will be \(-(10)^{19}\).

But how will you get \(- (10)^{20}\)?
You need to select -1 and twenty 10s but you cannot select 21 numbers. You cannot have the product as negative \(10^{20}\).
Hence (E) is not possible.


E is possible if you select 10 odd number of times and -10 the remaining number of times. For example:

\(10^1*(-10)^{19}=-10^{20}\) or \(10^3*(-10)^{17}=-10^{20}\) or \(10^5*(-10)^{15}=-10^{20}\)... or \(10^{19}*(-10)^{1}=-10^{20}\).

So, the answer is E.



Hi Bunuel : can you please help me understand how -10^20 is negative integer & -2^2 is a positive integer , although they have the negative signs and even exponents. I’m little confused here.

Posted from my mobile device


-10^20, which is negative, is different from (-10)^20, which is positive.

-10^20 is essentially -1*10^20 = negative*positive = negative.

P.S. -2^2 is also negative: -2^2 = -4, while (-2)^2 = 4.
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