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# M28-14

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 01:28
Expert's post
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Difficulty:

55% (hard)

Question Stats:

57% (01:42) correct 43% (01:19) wrong based on 46 sessions

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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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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 $$m=-20$$.

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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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21 Oct 2014, 01:10
tangt16 wrote:
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?

No. 381m = 380m + m.
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25 Dec 2015, 01:41
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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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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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29 Aug 2016, 05:26
siddhanthsivaraman wrote:
Hi Bunuel,
if this is the product of two consecutive negative integers,then isnt m=-19 instead of -20?(m+1=-20)

No.

m(m + 1) = 380;
(-20)(-20 + 1) =380;
(-20)(-19) = 380.
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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 $$m=-20$$.

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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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 $$m=-20$$.

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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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

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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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03 Jul 2017, 23:11
sidagar wrote:
hi brunel could you pls clear how to elaborate:

m3−m = m(m+1)(m−1)

$$m^3 - m$$

Factor out m: $$m(m^2-1)$$

Apply $$a^2 - b^2=(a+b)(a-b)$$ to $$m^2 -1$$: $$m(m^2-1)=m(m+1)(m-1)$$.

Hope it's clear.
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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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28 Sep 2017, 08:06
pars3008 wrote:
Is this approach correct?

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

Then plug in numbers and check.

Yes, you can do this way too.
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Re: M28-14   [#permalink] 28 Sep 2017, 08:06
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# M28-14

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