If m and n are Integers, What is the Value of m + n? GMAT Data Sufficiency

Question: 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

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: A

Solution and Explanation:
Approach Solution 1:
Given: m and n are integers

Statement 1: (x + m)(x + n) = x² + 5x + mn and x ≠ 0.
Using 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.

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.

Approach Solution 2:
1) We know that (x+m)(x+n)=x2+5x+mn and x≠0.
x^2+mx+nx+mn= x^2+5x+mn
x(m+n)= 5x

Since x≠0, we can divide both sides of the above equation by x.
m+n=5. Thus, the answer to the original question is a unique value.⟹ Sufficient

2) We know that mn= 4 and can test possible cases. If m= 2 and n= 2, then m+n= 4. However, if m= 4 and n= 1. Then m+n=5. Thus, we can't get a unique value to answer the original question.⟹ Insufficient

Approach Solution 3:
(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

“If m and n are integers, what is the value of m + n ?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official 2021". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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