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Math Expert V
Joined: 02 Sep 2009
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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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Math Expert V
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Posts: 58465

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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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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?
Math Expert V
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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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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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Hi Bunuel,
if this is the product of two consecutive negative integers,then isnt m=-19 instead of -20?(m+1=-20)
Math Expert V
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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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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?
Math Expert V
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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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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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hi brunel could you pls clear how to elaborate:

m3−m = m(m+1)(m−1)
Math Expert V
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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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Is this approach correct?

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

Then plug in numbers and check.
Math Expert V
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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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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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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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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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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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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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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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