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26 May 2017, 04:42
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If \(k\) and \(n\) are positive integers, is \(k\) a factor of \(n\)?
(1) \(k^{2} + 4k - 5 = 0\)
(2) \(n = 3\)
(1) \(k^{2} + 4k - 5 = 0\)
(2) \(n = 3\)
26 May 2017, 04:42
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Official Solution:
In the original condition, there are 2 variables \((n, k)\). In order to match the number of variables to the number of equations, there must be 2 equations. Therefore, C is most likely to be the answer. By solving con 1) and con 2), \(k^{2} + 4k - 5 = 0, (k + 5)(k - 1) = 0\), then you get \(k = -5\), 1. If \(k > 0\), you get \(k = 1\), and if \(n = 3\), you get \(3 = 1*3\), then you get \(n = k*3\). \(K\) becomes the factor of \(n\), hence yes, it is sufficient. Therefore, the answer is C. However, this is an integer question, one of the key questions, so you can apply "CMT 4(A: if you get C too easily, consider A or B)".
In the case of con 1), \(k^{2} + 4k - 5 = 0, (k+5)(k-1) = 0\), you get \(k = -5, 1,\) so \(k > 0\), then \(k = 1\). 1 is the factor of all positive integers, hence yes, it is sufficient.
In the case of con 2), if \(n = k = 3\), it is yes. If \(n = 3\), \(k = 5\), no, it is not sufficient. The answer is A.
Answer: A
In the original condition, there are 2 variables \((n, k)\). In order to match the number of variables to the number of equations, there must be 2 equations. Therefore, C is most likely to be the answer. By solving con 1) and con 2), \(k^{2} + 4k - 5 = 0, (k + 5)(k - 1) = 0\), then you get \(k = -5\), 1. If \(k > 0\), you get \(k = 1\), and if \(n = 3\), you get \(3 = 1*3\), then you get \(n = k*3\). \(K\) becomes the factor of \(n\), hence yes, it is sufficient. Therefore, the answer is C. However, this is an integer question, one of the key questions, so you can apply "CMT 4(A: if you get C too easily, consider A or B)".
In the case of con 1), \(k^{2} + 4k - 5 = 0, (k+5)(k-1) = 0\), you get \(k = -5, 1,\) so \(k > 0\), then \(k = 1\). 1 is the factor of all positive integers, hence yes, it is sufficient.
In the case of con 2), if \(n = k = 3\), it is yes. If \(n = 3\), \(k = 5\), no, it is not sufficient. The answer is A.
Answer: A
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