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06 Apr 2023, 04:23
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If \(m\) and \(n\) are positive integers, is \(4^m*(\frac{1}{3})^n < 1\)?
(1) \(n = 2m \)
(2) \(n = 4\)
(1) \(n = 2m \)
(2) \(n = 4\)
06 Apr 2023, 04:23
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Official Solution:
If \(m\) and \(n\) are positive integers, is \(4^m*(\frac{1}{3})^n < 1\)?
First, let's simplify the question:
Is \(4^m*\frac{1}{3^n} < 1\)?
Is \(4^m < 3^n\)?
(1) \(n = 2m \)
Substituting \(n = 2m\) into the question, it becomes:
Is \(4^m < 3^{2m}\)?
Is \(4^m < 9^{m}\)?
Since given that \(m\) is a positive integer, the answer to this question is YES. Sufficient.
(2) \(n = 4\)
Substituting \(n = 4\) into the question, it becomes:
Is \(4^m < 3^4\)?
Is \(4^m < 81\)?
If \(m=1\), the answer is YES. However, if \(m = 100\), the answer is NO. Not sufficient.
Answer: A
If \(m\) and \(n\) are positive integers, is \(4^m*(\frac{1}{3})^n < 1\)?
First, let's simplify the question:
Is \(4^m*\frac{1}{3^n} < 1\)?
Is \(4^m < 3^n\)?
(1) \(n = 2m \)
Substituting \(n = 2m\) into the question, it becomes:
Is \(4^m < 3^{2m}\)?
Is \(4^m < 9^{m}\)?
Since given that \(m\) is a positive integer, the answer to this question is YES. Sufficient.
(2) \(n = 4\)
Substituting \(n = 4\) into the question, it becomes:
Is \(4^m < 3^4\)?
Is \(4^m < 81\)?
If \(m=1\), the answer is YES. However, if \(m = 100\), the answer is NO. Not sufficient.
Answer: A
