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30 Dec 2023, 23:39
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If \(m\) and \(n\) are integers, how many pairs of \((m, n)\) are there such that \(3m + 2n = mn\)?
A. 4
B. 5
C. 6
D. 8
E. 10
A. 4
B. 5
C. 6
D. 8
E. 10
30 Dec 2023, 23:39
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Official Solution:
If \(m\) and \(n\) are integers, how many pairs of \((m, n)\) are there such that \(3m + 2n = mn\)?
A. 4
B. 5
C. 6
D. 8
E. 10
Arguably the best way to solve this question is to manipulate \(3m + 2n = mn\) such that we can factor it into \((m + a)(n + b) = integer\). Then, benefiting from the fact that \(m\) and \(n\) are integers, we'd be able to find the sets of \((m, n)\) satisfying the equation.
\(3m + 2n = mn\);
\(3m - mn + 2n = 0[\);
\(m(3 - n) + 2n = 0\).
To further factor \((3 - n)\), add -6 to both sides:
\(m(3 - n) - 6 + 2n= -6\);
\(m(3 - n) - 2(3 - n) = -6\);
\((3 - n)(m - 2) = -6\).
Now, the question essentially boils down to finding how many ways -6 can be written as the product of two integers, \((3 - n)\) and \((m - 2)\):
\(-6*1\)
\(-3*2\)
\(-2*3\)
\(-1*6\)
\(1*(-6)\)
\(2*(-3)\)
\(3*(-2)\)
\(6*(-1)\)
Therefore, there are eight sets of integers \((m, n)\) satisfying \((3 - n)(m - 2) = -6\). To find the exact values, though unnecessary, we equate \((3 - n)\) and \((m - 2)\) to each pair of factors above.
Answer: D
If \(m\) and \(n\) are integers, how many pairs of \((m, n)\) are there such that \(3m + 2n = mn\)?
A. 4
B. 5
C. 6
D. 8
E. 10
Arguably the best way to solve this question is to manipulate \(3m + 2n = mn\) such that we can factor it into \((m + a)(n + b) = integer\). Then, benefiting from the fact that \(m\) and \(n\) are integers, we'd be able to find the sets of \((m, n)\) satisfying the equation.
\(3m + 2n = mn\);
\(3m - mn + 2n = 0[\);
\(m(3 - n) + 2n = 0\).
To further factor \((3 - n)\), add -6 to both sides:
\(m(3 - n) - 6 + 2n= -6\);
\(m(3 - n) - 2(3 - n) = -6\);
\((3 - n)(m - 2) = -6\).
Now, the question essentially boils down to finding how many ways -6 can be written as the product of two integers, \((3 - n)\) and \((m - 2)\):
\(-6*1\)
\(-3*2\)
\(-2*3\)
\(-1*6\)
\(1*(-6)\)
\(2*(-3)\)
\(3*(-2)\)
\(6*(-1)\)
Therefore, there are eight sets of integers \((m, n)\) satisfying \((3 - n)(m - 2) = -6\). To find the exact values, though unnecessary, we equate \((3 - n)\) and \((m - 2)\) to each pair of factors above.
Answer: D
