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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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05 Jan 2018, 08:31

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dave13

Bunuel

catty2004

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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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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catty2004

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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13 Dec 2020, 05:03

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Bunuel, VeritasKarishma - what would be the answer to this question if repetition was not allowed? Would it be 0?

KarishmaB

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14 Dec 2020, 22:26

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davidbeckham

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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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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Updated on: 15 Feb 2022, 06:12

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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KarishmaB

catty2004

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

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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 $\left(-10\right)^x\cdot\left(10\right)^{20-x}\ =\ -10^{20}$

KarishmaB

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AbhiR3

KarishmaB

catty2004

But i'm still not sure, why the other answer choices are eliminated and why E is the least possible answer?? >_<

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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dave13

Bunuel

catty2004

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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Bunuel

VeritasPrepKarishma

catty2004

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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MasteringGMAT

Bunuel

VeritasPrepKarishma

-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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I originally got tripped up because I didn't read "repetitions allowed".

We want the smallest number possible.
We have to choose 20 numbers.
Repetitions are allowed

If we multiple -10 by itself 20 times, it will result in a positive number, the opposite of what we want.

Multiple -10 by itself an odd number of times:
-10^19

Now multiple -10^19 by positive 10 to keep the number negative:
-10^19 x -10 = (-10)^20 WRONG
-10^19 x 10 = -(10)^20

Keeping the - sign outside the parentheses keeps the final result negative.

E

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catty2004

Here i explain it to you all in a simple way
They tell -10 to 10 its 21 integers we need only 20
Now see if we choose 0 its give us 0 so its eliminate
A and B are eliminate beacasue see their power its even and we know even power give us positive number that is more so A and B eliminate
About D and E here we see if we do their real value its give in minus but here we need to choose the Greater Minus value like not -2 its like -5
So here least Possible value is gonna BY E -(10)^20

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Quote:

Interesting question.

Two things that can help ground us is that:

1) The correct answer will be negative. This is because if, out of the 20 integers randomly chosen with repititions allowed, 19 were positive and 1 was negative, the product would be negative. Since a negative total value is clearly possible given the parameters, the LEAST POSSIBLE VALUE has to be negative as well. We may not know what the exact value is immediately, but we know it has to be negative.

2) If a minus sign is INSIDE the bracket, an EVEN exponent will make the total value positive.

We can quickly get rid of A, B, and C therefore.

The only two options that are sort of close are (D) and (E), since the correct answer choice has to be negative.

What's interesting about these two options is where the minus sign is positioned (i.e. outside the bracket). Carefully looking at the two, we can see that (E) is less than (D). If it helps, think of the value of (10)^19 and (10)^20. If you were to then attach a negative sign to each value, which would be less?

(E) is your answer.