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M28-14

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M28-14 [#permalink]

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If \(m\) is a negative integer and \(m^3 + 380 = 381m\), then what is the value of \(m\)?

A. \(-21\)
B. \(-20\)
C. \(-19\)
D. \(-1\)
E. \(None \ of \ the \ above\)
[Reveal] Spoiler: OA

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New post 16 Sep 2014, 01:28
Official Solution:

If \(m\) is a negative integer and \(m^3 + 380 = 381m\), then what is the value of \(m\)?

A. \(-21\)
B. \(-20\)
C. \(-19\)
D. \(-1\)
E. \(None \ of \ the \ above\)


Given \(m^3 + 380 = 380m+m\).

Re-arrange: \(m^3-m= 380m-380\).

\(m(m+1)(m-1)=380(m-1)\). Since \(m\) is a negative integer, then \(m-1\neq{0}\) and we can safely reduce by \(m-1\) to get \(m(m+1)=380\).

So, we have that 380 is the product of two consecutive negative integers: \(380=-20*(-19)\), hence \(m=-20\).


Answer: B
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Re: M28-14 [#permalink]

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New post 20 Oct 2014, 22:44
If the question statement is "m is a negative integer and m^3 + 380 = 381m", how did you derive that "given m^3 + 380 = 380m+m." Is this a typo and there is a +m missing?

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Re: M28-14 [#permalink]

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not having that algebraic gift i did it the following way: the stem simplifies to m(m^2-381)= -380, where we know that m is -ve therefore (m^2-381) should be +ve for the product to be a -ve number.

-1 and -19 rule out because they would leave a -ve results int he brackets and hence give +ve product which cannot be true. therefore we need to brute force options -20 and -21. start with -20 and you get the desired result
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Re: M28-14 [#permalink]

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New post 29 Aug 2016, 05:14
Hi Bunuel,
if this is the product of two consecutive negative integers,then isnt m=-19 instead of -20?(m+1=-20)

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M28-14 [#permalink]

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New post 17 Jun 2017, 07:51
Bunuel wrote:
Official Solution:

If \(m\) is a negative integer and \(m^3 + 380 = 381m\), then what is the value of \(m\)?

A. \(-21\)
B. \(-20\)
C. \(-19\)
D. \(-1\)
E. \(None \ of \ the \ above\)


Given \(m^3 + 380 = 381m\).

Re-arrange: \(m^3-m= 380m-380\).

\(m(m+1)(m-1)=380(m-1)\). Since \(m\) is a negative integer, then \(m-1\neq{0}\) and we can safely reduce by \(m-1\) to get \(m(m+1)=380\).

So, we have that 380 is the product of two consecutive negative integers: \(380=-20*(-19)\), hence \(m=-20\).


Answer: B


Great explanation as always. However, I approached it differently.

\(m^3 + 380 = 381m\)

\(m(m^2-381)=-380\) then \(m^2-380=\frac{-380}{m}\)
next \(m^2=381-\frac{380}{m}\), and now we plug in the answer choices and only (C) -19 gives us the right answer

What do you think Bunuel?

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Re: M28-14 [#permalink]

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New post 17 Jun 2017, 10:50
MikeMighty wrote:
Bunuel wrote:
Official Solution:

If \(m\) is a negative integer and \(m^3 + 380 = 381m\), then what is the value of \(m\)?

A. \(-21\)
B. \(-20\)
C. \(-19\)
D. \(-1\)
E. \(None \ of \ the \ above\)


Given \(m^3 + 380 = 381m\).

Re-arrange: \(m^3-m= 380m-380\).

\(m(m+1)(m-1)=380(m-1)\). Since \(m\) is a negative integer, then \(m-1\neq{0}\) and we can safely reduce by \(m-1\) to get \(m(m+1)=380\).

So, we have that 380 is the product of two consecutive negative integers: \(380=-20*(-19)\), hence \(m=-20\).


Answer: B


Great explanation as always. However, I approached it differently.

\(m^3 + 380 = 381m\)

\(m(m^2-381)=-380\) then \(m^2-380=\frac{-380}{m}\)
next \(m^2=381-\frac{380}{m}\), and now we plug in the answer choices and only (C) -19 gives us the right answer

What do you think Bunuel?


Yes, you can use plug-in method to solve this question.
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Re: M28-14 [#permalink]

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New post 18 Jun 2017, 06:56
MikeMighty wrote:

Great explanation as always. However, I approached it differently.

\(m^3 + 380 = 381m\)

\(m(m^2-381)=-380\) then \(m^2-380=\frac{-380}{m}\)
next \(m^2=381-\frac{380}{m}\), and now we plug in the answer choices and only (C) -19 gives us the right answer

What do you think Bunuel?


Hi Mike,

I would like to highlight that the correct answer is B not C. The right answer is -20.

I did the same like you till a point and used other reasoning. Usign sense of numbers and memorizing some square of integers a re perfect. I will elaborate.

\(m(m^2-381)=-380\).........I need two numbers that gives -380

m^2 can't me less than or equal to 381 otherwise, it will be as follows:

Case 1:

Negative ( Positive less 381 - 381) = positive value .......Incorrect

Case 2:

Negative ( Positive equal to 381 - 381) = 0 .......Incorrect


D) -1

This will give you case 1......Eliminate D

C ) -19

m^2 = 381...So it is case 2.....Eliminate C

B) -20

-20 ( 400 -381)= -20 * 19 = -380.........Correct

Answer B

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Re: M28-14 [#permalink]

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New post 03 Jul 2017, 22:55
hi brunel could you pls clear how to elaborate:

m3−m = m(m+1)(m−1)

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New post 03 Jul 2017, 23:11

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Re: M28-14 [#permalink]

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New post 28 Sep 2017, 08:04
Is this approach correct?

Divide both side by M
m^2+ 380/m=381

Then plug in numbers and check.

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New post 28 Sep 2017, 08:06

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Re: M28-14   [#permalink] 28 Sep 2017, 08:06
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