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# Is the product of three consecutive integers negative?

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Math Expert
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Is the product of three consecutive integers negative?  [#permalink]

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28 Feb 2014, 04:14
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95% (hard)

Question Stats:

43% (01:06) correct 57% (01:15) wrong based on 218 sessions

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Is the product of three consecutive integers negative?

(1) The sum of the integers is equal to the product of the integers.

(2) At least one of the integers is negative.

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Re: Is the product of three consecutive integers negative?  [#permalink]

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28 Feb 2014, 04:14
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SOLUTION

Is the product of three consecutive integers negative?

(1) The sum of the integers is equal to the product of the integers. Let the three consecutive integers be $$(x-1)$$, $$x$$, and $$x+1$$, so we are given that $$(x-1)+x+(x+1)=(x-1)x(x+1)$$ --> $$3x=x(x^2-1)$$ --> $$x(x^2-4)=0$$ --> $$x=-2$$, $$x=0$$, or $$x=2$$. The set can be: {-3, -2, -1}, {-1, 0, 1}, or {1, 2, 3}. Not sufficient.

(2) At least one of the integers is negative.

We can have three cases:
(i) All three integers are negative. In this case the product will obviously be negative.
(ii) Two of the integers are negative: {-2, -1, 0}. In this case the product will be zero.
(iii) Only one of the integers is negative: {-1, 0, 1}. In this case the product will be zero.

Not sufficient.

(1)+(2) We still can have 2 sets: {-3, -2, -1} and {-1, 0, 1}. So, the product can be negative as well as zero. Not sufficient.

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Re: Is the product of three consecutive integers negative?  [#permalink]

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28 Feb 2014, 06:38
1
Consider consecutive integers as
a-1, a, a+1
1. According to statement 1 ,
Sum = product
so 3a= a ( a^2 -1) solving we get a=0, a=-2 , +2
three values of a so ,
integers are -1, 0, 1 -3,-2,-1, and 1, 2 , 3
as we cannot answer the target question with certainty ..Statement is insufficient
2. Second statement : clearly insufficient , as when 1 is negative = product is -ve
2 are -ve = product is +ve
So 2nd statement is also insufficient

Combining both we get consecutive integers as : -1, 0 , 1 ===product is zero
-3, -2, -1 ===product is -ve

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Re: Is the product of three consecutive integers negative?  [#permalink]

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28 Feb 2014, 07:09

Statement 1 is insufficient for example 1,2,3 have product and sum and product of 6
While -1,-2,-3 have a sum and product of -6 so there are two possibilities.

Statement 2 is insufficient because at least one negative means that there could be 1,2 or 3 numbers negative and each will have a different outcome and in the case of -1,0,1 the product is 0.

However, statement 1 and 2 taken together mean that the three numbers have to be negative and have a sum and product that is equal . So yes "C" means the product is negative !!

What is OA !!
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Re: Is the product of three consecutive integers negative?  [#permalink]

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28 Feb 2014, 15:07
1
Statement 1:
can be satisfied with (1,2,3) sum =6, product = 6
can be satisfied with (-1,-2,-3) sum = -6, product = -6

Insufficient

Statement 2: clearly insufficient

1 and 2:
can be satisfied with (-1,-2,-3)
can be satisfied with (-1,0,1) product is not negative

E
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Re: Is the product of three consecutive integers negative?  [#permalink]

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03 Mar 2014, 12:22
1
SOLUTION

Is the product of three consecutive integers negative?

(1) The sum of the integers is equal to the product of the integers. Let the three consecutive integers be $$(x-1)$$, $$x$$, and $$x+1$$, so we are given that $$(x-1)+x+(x+1)=(x-1)x(x+1)$$ --> $$3x=x(x^2-1)$$ --> $$x(x^2-4)=0$$ --> $$x=-2$$, $$x=0$$, or $$x=2$$. The set can be: {-3, -2, -1}, {-1, 0, 1}, or {1, 2, 3}. Not sufficient.

(2) At least one of the integers is negative.

We can have three cases:
(i) All three integers are negative. In this case the product will obviously be negative.
(ii) Two of the integers are negative: {-2, -1, 0}. In this case the product will be zero.
(iii) Only one of the integers is negative: {-1, 0, 1}. In this case the product will be zero.

Not sufficient.

(1)+(2) We still can have 2 sets: {-3, -2, -1} and {-1, 0, 1}. So, the product can be negative as well as zero. Not sufficient.

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Re: Is the product of three consecutive integers negative?  [#permalink]

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03 Mar 2014, 20:54
great example of why you can't blindly reduce equations [e.g. (x-1)x(x+1)=3x --> (x-1)(x+1)=3]
thanks Bunuel.
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Is the product of three consecutive integers negative?  [#permalink]

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08 Sep 2015, 07:24
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.

Is the product of three consecutive integers negative?

(1) The sum of the integers is equal to the product of the integers.

(2) At least one of the integers is negative.

In the original condition we have three consecutive integers thus we just need to know the first term, thus we have 1 variable. Since we need 1 equation, D is likely the answer.

In case of 1), for (-3,-2,-1) the answer is yes, for (-1,0,1) the answer is no, and same for (1,2,3). The answer is not unique, therefore it is not sufficient.
In case of 2), for (-1,0,1) the answer is no, and for (-3,-2,-1) the answer is yes. Thus the condition is not sufficient.
Using both 1) & 2) together, for (-1,0,1) the answer is no, for (-3,-2,-1) the answer is yes. therefore the conditions are not sufficient. The answer is E.
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Re: Is the product of three consecutive integers negative?  [#permalink]

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18 Jun 2018, 12:03
Hi Bunuel , great question - but can you please explain why we cannot blindly cancel upon the "x's" on both sides of the equation you used ( i.e. (x−1)x(x+1)(x−1)+x+(x+1)=(x−1)x(x+1) --> 3x )

I have noticed you not cancelling out such scenarios in multiple questions (including not drawing direct conclusions in ratio based questions as well) - any reason why? and any guidance on when not to cancel out such cases??

TIA!
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Re: Is the product of three consecutive integers negative?  [#permalink]

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18 Jun 2018, 14:03
NidSha wrote:
Hi Bunuel , great question - but can you please explain why we cannot blindly cancel upon the "x's" on both sides of the equation you used ( i.e. (x−1)x(x+1)(x−1)+x+(x+1)=(x−1)x(x+1) --> 3x )

I have noticed you not cancelling out such scenarios in multiple questions (including not drawing direct conclusions in ratio based questions as well) - any reason why? and any guidance on when not to cancel out such cases??

TIA!

Because that cancels out one result. You will notice that if you cut that x out, you will not get X=0 and thus believe that there is only one answer.

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Re: Is the product of three consecutive integers negative?  [#permalink]

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18 Jun 2018, 14:49
I.
x+x+1+x+2=x(x+1)(x+2)
3(x+1)=x(x+1)(x+2)
x=-1 then cancel x+1
3=x(x+2)
x=1, -3
1, 2, 3 Product is positive
-1, 0, 1 Product is zero
-3, -2, -1 Product is negative
Insufficient

II.
Insufficient

I and II.
Insufficient

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Re: Is the product of three consecutive integers negative?  [#permalink]

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18 Jun 2018, 17:33
Antreev - so should i generalize that each time we are trying to find solutions for an unknown variable, such cases should not be cancelled ?

Though this confuses me - don't we usually arrive to such equations to find possible solutions for an unknown variable?
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Re: Is the product of three consecutive integers negative?  [#permalink]

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18 Jun 2018, 20:58
1
NidSha wrote:
Antreev - so should i generalize that each time we are trying to find solutions for an unknown variable, such cases should not be cancelled ?

Though this confuses me - don't we usually arrive to such equations to find possible solutions for an unknown variable?

Hello Nidsha

If you could take particular example then it would be better to discuss question by question
It varies from question to question. Eg, if we have:
3x(x-1) = x(x+1)

then we cannot cancel x from both sides, because x could be '0' also, and cancellation by '0' (division by 0) is not permissible.

But in the above equation if we know that x is a non-zero number, then we can cancel x from both sides.

I would request you to take example of that particular question that you are talking about so we could discuss it in detail.
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Re: Is the product of three consecutive integers negative?  [#permalink]

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19 Jun 2018, 20:27
amanvermagmat - thank you so much for this explanation - i think i was hugely missing the '0' based fact/' exception while looking at such equations - explains it all now - thank you so much again!
Re: Is the product of three consecutive integers negative? &nbs [#permalink] 19 Jun 2018, 20:27
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