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
If m and n are integers, what is the value of m + n ?
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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.
(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.
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Given: m and n are integersAbeNeedsAnswers 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
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
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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
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
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For (1) , (x+m)(x+n) = x^2+x(m+n)+mnAbeNeedsAnswers 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
x^2+x(m+n)+mn = x^2+5x+mn
x(m+n) = 5x
m+n = 5
so, (1) is sufficient to answer the question
for (2) mn = 4 means
m n mn . m+n
1 . 4 . 4 . 5
2 . 2 . 4 . 4
4 . 1 . 4 . 5
so we get two values of 4 and 5 for m+n. clearly this is NOT sufficient.
Hence Answer is A.