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If \(m\) and \(n\) are integers, is \(mn > 0\)?
(1) \(\frac{m}{n}\) is an integer.
(2) \(|m| < |n|\)
(1) \(\frac{m}{n}\) is an integer.
(2) \(|m| < |n|\)
27 Apr 2023, 02:55
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Official Solution:
If \(m\) and \(n\) are integers, is \(mn > 0\)?
(1) \(\frac{m}{n}\) is an integer.
This statement is clearly insufficient. For example, consider \(m = 1\) and \(n = 1\), and \(m=1\) and \(n=-1\).
(2) \(|m| < |n|\)
This statement implies that \(n\) is further from 0 than \(m\). This statement is also clearly insufficient. For example, consider \(m = 1\) and \(n = 2\), and \(m=1\) and \(n=-2\).
(1)+(2) \(\frac{m}{n}\) being an integer implies that \(m\) is a multiple of \(n\), and \(|m| < |n|\) implies that \(m\) is smaller in magnitude than \(n\). The only way for \(m\) to be a multiple of \(n\) and have a smaller magnitude than \(n\) is if \(m\) is 0. Thus, \(m = 0\), which provides a NO answer to the question of whether \(mn > 0\). Sufficient.
Answer: C
If \(m\) and \(n\) are integers, is \(mn > 0\)?
(1) \(\frac{m}{n}\) is an integer.
This statement is clearly insufficient. For example, consider \(m = 1\) and \(n = 1\), and \(m=1\) and \(n=-1\).
(2) \(|m| < |n|\)
This statement implies that \(n\) is further from 0 than \(m\). This statement is also clearly insufficient. For example, consider \(m = 1\) and \(n = 2\), and \(m=1\) and \(n=-2\).
(1)+(2) \(\frac{m}{n}\) being an integer implies that \(m\) is a multiple of \(n\), and \(|m| < |n|\) implies that \(m\) is smaller in magnitude than \(n\). The only way for \(m\) to be a multiple of \(n\) and have a smaller magnitude than \(n\) is if \(m\) is 0. Thus, \(m = 0\), which provides a NO answer to the question of whether \(mn > 0\). Sufficient.
Answer: C
