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# M28-14

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Math Expert
Joined: 02 Sep 2009
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M28-14 [#permalink]

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16 Sep 2014, 00:28
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Difficulty:

55% (hard)

Question Stats:

56% (01:37) correct 44% (01:18) wrong based on 57 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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Re M28-14 [#permalink]

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16 Sep 2014, 00:28
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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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20 Oct 2014, 21:44
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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21 Oct 2014, 00:10
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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Re: M28-14 [#permalink]

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25 Dec 2015, 00:41
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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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Re: M28-14 [#permalink]

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29 Aug 2016, 04:14
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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Re: M28-14 [#permalink]

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29 Aug 2016, 04:26
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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M28-14 [#permalink]

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17 Jun 2017, 06:51
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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17 Jun 2017, 09:50
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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18 Jun 2017, 05:56
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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03 Jul 2017, 21:55
hi brunel could you pls clear how to elaborate:

m3−m = m(m+1)(m−1)
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Re: M28-14 [#permalink]

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03 Jul 2017, 22:11
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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Re: M28-14 [#permalink]

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28 Sep 2017, 07:04
Is this approach correct?

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

Then plug in numbers and check.
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Posts: 43363
Re: M28-14 [#permalink]

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28 Sep 2017, 07:06
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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M28-14 [#permalink]

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11 Nov 2017, 16:34
I did this a bit differently though not sure if that was fluke or not -

On the Right side we have -

381m = which would clearly be a multiple of m

On the left we have-
$$m^3$$ + 380. Now $$m^3$$ is clearly a multiple of m too so 380 must be a multiple of m too.

Now test for are -20, -19 and -1.
-1 and -19 dont work when used in the equation.
-20 works. Hence B.

Is this the right approach?
M28-14   [#permalink] 11 Nov 2017, 16:34
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# M28-14

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