If m and n are integers, what is the value of m + n ?

If you expand the left side of Statement 1, we have

(x + m)(x + n) = x^2 + 5x + mn
x^2 + (m+n)x + mn = x^2 + 5x + mn

and now most of the terms can be subtracted from both sides, leaving us with

(m + n)x = 5x

and dividing by the nonzero x, we find m+n = 5, so Statement 1 is sufficient.

Statement 2 is not sufficient, because there are a few ways mn = 4 can be true for two integers m and n, and we can get different values of m+n. For example, we might have m=4 and n=1 or we might have m=-4 and n=-1.

AbeNeedsAnswers wrote:If m and n are integers, what is the value of m + n ?

(1) (x + m)(x + n) = x^2 + 5x + mn and x ≠ 0.
(2) mn = 4

A

Source: Official Guide 2020

Given: m and n are integers

Target question: What is the value of m + n ?

Statement 1: (x + m)(x + n) = x² + 5x + mn and x ≠ 0.
Use FOIL to expand the left side: x² + nx + mx + mn = x² + 5x + mn
Factor the two middle terms: x² + x(n + m) + mn = x² + 5x + mn
At this point, we should see that m+n = 5
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

If you're not convinced, we can take a few more steps.
Take: x² + x(n + m) + mn = x² + 5x + mn
Subtract x² from both sides: x(n + m) + mn = 5x + mn
Subtract mn from both sides: x(n + m) = 5x
Divide both sides by x to get: n + m = 5
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: mn = 4
There are many values of m and n that satisfy statement 2. Here are two:
Case a: m = 4 and n = 1. In this case, the answer to the target question is m + n = 4 + 1 = 5
Case b: m = 2 and n = 2. In this case, the answer to the target question is m + n = 2 + 2 = 4
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

Hi All,

We're told that M and N are INTEGERS. We're asked for the value of M+N. This question can be solved with a mix of Algebra and TESTing VALUES.

(1) (X + M)(X + N) = X^2 + 5X + (M)(N) and X ≠ 0.

The equation in Fact 1 should get us thinking about a Quadratic and how we can F.O.I.L. the parentheses and get the given result. Notice that the 'middle' term is "+5X"; this provides a huge clue about how M and N relate to one another. Since the SUM of M and N must be +5, you can stop right here (since the question asks for that exact value). If you didn't immediately recognize that though, you can run a few TESTs and see what happens...

IF....
M=1 and N=4, then (X+1)(X+4) = X^2 +5X + 4 and the answer to the question is 5.
M=2 and N=3, then (X+2)(X+3) = X^2 +5X + 6 and the answer to the question is 5.
M=6 and N= -1, then (X+6)(X-1) = X^2 +5X - 6 and the answer to the question is 5.
Etc.
The answer to the question is ALWAYS 5.
Fact 1 is SUFFICIENT

(2) (M)(N) = 4

With the equation in Fact 2, we can come up with a couple of simple examples...
IF....
M=1 and N=4, then the answer to the question is 5.
M=2 and N=2, then the answer to the question is 4.
Fact 2 is INSUFFICIENT

Final Answer: A

GMAT assassins aren't born, they're made,
Rich