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16 Sep 2014, 00:50
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If \(a\) and \(b\) are positive integers, is \(a^b > b^a\) ?
(1) \(a > b > 1\).
(2) \(a = b + 1\).
(1) \(a > b > 1\).
(2) \(a = b + 1\).
16 Sep 2014, 00:50
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Official Solution:
If \(a\) and \(b\) are positive integers, is \(a^b > b^a\) ?
Testing numbers is arguably the most effective strategy for this problem.
(1) \(a > b > 1\).
When \(a\) is much larger than \(b\), it's easy to obtain a NO answer:
If \(a = 10\) and \(b = 2\), then \(a^b = 10^2 = 100\), and \(b^a = 2^{10} = 1024\). Hence, in this case, \(a^b < b^a\), and we'd have a NO answer.
To attempt to get a YES answer for this inequality, we should try picking numbers where \(a\) and \(b\) are as small as possible:
If \(a = 3\) and \(b = 2\), then \(a^b = 3^2 = 9\), and \(b^a = 2^3 = 8\). Hence, in this case, \(a^b > b^a\), and we'd have a YES answer.
Not sufficient.
(2) \(a = b + 1\).
When \(a\) and \(b\) are positive integers and \(a = b + 1\), then for sufficiently large numbers, \(a^b < b^a\) always holds, giving a NO answer. For example, \(5^{4} < 4^{5}\), \(11^{10} < 10^{11}\), \(110^{109} < 109^{110}\), and so on.
However, there are exceptions when \(a\) and \(b\) are very small:
If \(a = 2\) and \(b = 1\), then \(a^b = 2^1 = 2\), and \(b^a = 1^2 = 1\). Hence, in this case, \(a^b > b^a\), and we'd have a YES answer.
If \(a = 3\) and \(b = 2\), then \(a^b = 3^2 = 9\), and \(b^a = 2^3 = 8\). Hence, in this case, \(a^b > b^a\), and we'd have a YES answer.
Not sufficient.
(1)+(2) When combining the statements, we have \(a = b + 1\) and \(b > 1\). To find a NO answer, we can test with larger values of \(a\) and \(b\). For example, \(a = 4\) and \(b = 3\) give a NO answer. However, for the smallest values, like \(a = 3\) and \(b = 2\), we get a YES answer. Not sufficient.
Answer: E
If \(a\) and \(b\) are positive integers, is \(a^b > b^a\) ?
Testing numbers is arguably the most effective strategy for this problem.
(1) \(a > b > 1\).
When \(a\) is much larger than \(b\), it's easy to obtain a NO answer:
If \(a = 10\) and \(b = 2\), then \(a^b = 10^2 = 100\), and \(b^a = 2^{10} = 1024\). Hence, in this case, \(a^b < b^a\), and we'd have a NO answer.
To attempt to get a YES answer for this inequality, we should try picking numbers where \(a\) and \(b\) are as small as possible:
If \(a = 3\) and \(b = 2\), then \(a^b = 3^2 = 9\), and \(b^a = 2^3 = 8\). Hence, in this case, \(a^b > b^a\), and we'd have a YES answer.
Not sufficient.
(2) \(a = b + 1\).
When \(a\) and \(b\) are positive integers and \(a = b + 1\), then for sufficiently large numbers, \(a^b < b^a\) always holds, giving a NO answer. For example, \(5^{4} < 4^{5}\), \(11^{10} < 10^{11}\), \(110^{109} < 109^{110}\), and so on.
However, there are exceptions when \(a\) and \(b\) are very small:
If \(a = 2\) and \(b = 1\), then \(a^b = 2^1 = 2\), and \(b^a = 1^2 = 1\). Hence, in this case, \(a^b > b^a\), and we'd have a YES answer.
If \(a = 3\) and \(b = 2\), then \(a^b = 3^2 = 9\), and \(b^a = 2^3 = 8\). Hence, in this case, \(a^b > b^a\), and we'd have a YES answer.
Not sufficient.
(1)+(2) When combining the statements, we have \(a = b + 1\) and \(b > 1\). To find a NO answer, we can test with larger values of \(a\) and \(b\). For example, \(a = 4\) and \(b = 3\) give a NO answer. However, for the smallest values, like \(a = 3\) and \(b = 2\), we get a YES answer. Not sufficient.
Answer: E
15 Dec 2019, 15:45
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I think this is a high-quality question and I agree with explanation.
