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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
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Difficulty:

75% (hard)

Question Stats:

54% (02:07) correct 46% (02:12) wrong based on 180 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. $$None \ of \ the \ above$$

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16 Sep 2014, 01:28
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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 $$m=-20$$.

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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
1
1
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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11 Nov 2017, 17:34
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?
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18 Mar 2018, 03:11
Bunuel wrote:
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. $$None \ of \ the \ above$$

HI Bunuel,

$$m^3 + 380 = 381m$$ ==> $$m^3-381m=-380$$

$$m(m^2-381)=-380$$==> m=-380 or $$(m^2-381)=-380$$

How to proceed further?
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30 Mar 2018, 07:11
NandishSS wrote:
Bunuel wrote:
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. $$None \ of \ the \ above$$

HI Bunuel,

$$m^3 + 380 = 381m$$ ==> $$m^3-381m=-380$$

$$m(m^2-381)=-380$$==> m=-380 or $$(m^2-381)=-380$$

How to proceed further?

HI GMATPrepNow / Brent

Can you pls help me with above query?
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30 Mar 2018, 08:06
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Top Contributor
NandishSS wrote:
NandishSS wrote:
Bunuel wrote:
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. $$None \ of \ the \ above$$

HI Bunuel,

$$m^3 + 380 = 381m$$ ==> $$m^3-381m=-380$$

$$m(m^2-381)=-380$$==> m=-380 or $$(m^2-381)=-380$$

How to proceed further?

HI GMATPrepNow / Brent

Can you pls help me with above query?

This can be solved with some TRICKY factoring.

Given: m³ - 381m + 380 = 0
Rewrite -381m as -m - 380m to get: m³ - m - 380m + 380 = 0
Factor IN PARTS to get: m(m² - 1) - 380(m - 1) = 0
Factor m² - 1 to get: m(m+1)(m-1) - 380(m-1) = 0
Collect "like" terms to get: (m-1)[ m(m+1) - 380] = 0
Simplify: (m-1)(m² + m - 380) = 0
Factor to get: (m-1)(m+20)(m-19) = 0

So, the possible m-values are 1, -20 and 19
The question tells us that m is NEGATIVE

So, m must equal -20

Cheers,
Brent
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27 May 2018, 14:25
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Bunuel wrote:
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. $$None \ of \ the \ above$$

Similar to plug-in method, but more efficient in my opinion is to use cyclicity.

Start with choice A. -21^3 will have 1 in the units digit. So the LHS (-1 + 380) has 9 in the units digit. But the RHS (381 * -21) has 1 in the units digit, so this cant be the answer.
Repeat this process for all options, and you'll see that only -20 passes the units digit test. At this point, -20 *might* be the answer, but we dont know for sure. You'll have to brute force it to confirm.
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24 Jun 2018, 23:36
The question is solvable much faster by substituting numbers.

As per MGMAT guidance, start with either (B) or (D), but it probably takes c.1 min, and is much easier than the explanation above.
Re: M28-14   [#permalink] 24 Jun 2018, 23:36

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# M28-14

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