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
_________________

No. 381m = 380m + m.
_________________

Hi Bunuel,
if this is the product of two consecutive negative integers,then isnt m=-19 instead of -20?(m+1=-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?

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

\(m^3 - m\)

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.
_________________

Yes, you can do this way too.
_________________

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?
_________________

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

Answer: B

Cheers,
Brent
_________________

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.