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16 Sep 2014, 00:17
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If \(m\) and \(n\) are positive integers, and \(x = 2^m3^n\), is \(m < n\) ?
(1) \(x\) is divisible by 144.
(2) \(x\) is not divisible by 648.
(1) \(x\) is divisible by 144.
(2) \(x\) is not divisible by 648.
16 Sep 2014, 00:17
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Official Solution:
If \(m\) and \(n\) are positive integers, and \(x = 2^m3^n\), is \(m < n\) ?
(1) \(x\) is divisible by 144.
Factoring 144, we get \(144 = 2^4 * 3^2\).
Hence, this statement implies that \(m ≥ 4\) and \(n ≥ 2\). However, without knowing the upper limits of \(m\) and \(n\), we cannot determine whether \(m < n\). Not sufficient.
(2) \(x\) is not divisible by 648.
Factoring 648, we get \(648 = 2^3 * 3^4\).
Hence, this statement implies either \(m < 3\) or \(n < 4\), or both. For instance, if \(m = 2\) and \(n = 10\), then \(m < n\). But if \(m = 10\) and \(n = 3\), then \(m > n\). Not sufficient.
(1)+(2) From (1) we know \(m ≥ 4\), so the condition \(m < 3\) from (2) cannot be true. Therefore, based on (2), \(n < 4\) must be true. This means \(m ≥ 4\) and \(n < 4\), leading to the conclusion that \(m > n\). Sufficient.
Answer: C
If \(m\) and \(n\) are positive integers, and \(x = 2^m3^n\), is \(m < n\) ?
(1) \(x\) is divisible by 144.
Factoring 144, we get \(144 = 2^4 * 3^2\).
Hence, this statement implies that \(m ≥ 4\) and \(n ≥ 2\). However, without knowing the upper limits of \(m\) and \(n\), we cannot determine whether \(m < n\). Not sufficient.
(2) \(x\) is not divisible by 648.
Factoring 648, we get \(648 = 2^3 * 3^4\).
Hence, this statement implies either \(m < 3\) or \(n < 4\), or both. For instance, if \(m = 2\) and \(n = 10\), then \(m < n\). But if \(m = 10\) and \(n = 3\), then \(m > n\). Not sufficient.
(1)+(2) From (1) we know \(m ≥ 4\), so the condition \(m < 3\) from (2) cannot be true. Therefore, based on (2), \(n < 4\) must be true. This means \(m ≥ 4\) and \(n < 4\), leading to the conclusion that \(m > n\). Sufficient.
Answer: C
General Discussion
18 Aug 2023, 10:05
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
