Bunuel wrote:
If m and n are positive integers, what is the value of m + n?
(1) \(5m - mn = 3\)
(2) \(m(m + 2n) + 4 = 13 - n^2\)
Given: m and n are positive integers Target question: What is the value of m + n? Statement 1: \(5m - mn = 3\) Factor the left side to get: \(m(5 - n) = 3\)
Since \(m\) and \(5-n\) must be
positive integers, there are only two possible cases to consider:
Case a: \(m = 3\) and \(5-n = 1\), which means \(n = 4\). In this case, the answer to the target question is
\(m + n = 3 + 4 = 7\)Case b: \(m = 1\) and \(5-n = 3\), which means \(n = 2\). In this case, the answer to the target question is
\(m + n = 1 + 2 = 3\)Since we can’t answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: \(m(m + 2n) + 4 = 13 - n^2\)Expand the left side to get: \(m^2 + 2mn + 4 = 13 - n^2\)
Rearrange to get: \(m^2 + 2mn + n^2 = 9\)
Factor the left side: \((m+n)^2 = 9\)
Since we are told m and n are \(positive\), it must be the case that
m + n = 3Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: B