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26 Apr 2023, 04:11
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If \(n\) is an integer such that \(\frac{1}{6} < \frac{1}{n-1} < \frac{1}{3}\), what is the value of \(n\)?
(1) \((n - 6)(n - 7) = 0\).
(2) \((n - 5)(n - 3) ≠ 0\).
(1) \((n - 6)(n - 7) = 0\).
(2) \((n - 5)(n - 3) ≠ 0\).
26 Apr 2023, 04:11
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Official Solution:
If \(n\) is an integer such that \(\frac{1}{6} < \frac{1}{n-1} < \frac{1}{3}\), what is the value of \(n\)?
Given that \(n\) is an integer, for the inequality \(\frac{1}{6} < \frac{1}{n-1} < \frac{1}{3}\) to hold true, \(\frac{1}{n-1}\) must be either \(\frac{1}{5}\) or \(\frac{1}{4}\). Hence, \(n\) must be either 6 or 5. The question essentially asks: if \(n\) is either 6 or 5, what is the value of \(n\)?
(1) \((n - 6)(n - 7) = 0\).
This statement implies that \(n\) is either \(6\) or \(7\). Therefore, \(n\) cannot be 5, and it must be 6. Sufficient.
(2) \((n - 5)(n - 3) ≠ 0\).
This statement implies that \(n\) is neither \(5\) nor \(3\). Since \(n\) is not 5, it must be 6. Sufficient
Answer: D
If \(n\) is an integer such that \(\frac{1}{6} < \frac{1}{n-1} < \frac{1}{3}\), what is the value of \(n\)?
Given that \(n\) is an integer, for the inequality \(\frac{1}{6} < \frac{1}{n-1} < \frac{1}{3}\) to hold true, \(\frac{1}{n-1}\) must be either \(\frac{1}{5}\) or \(\frac{1}{4}\). Hence, \(n\) must be either 6 or 5. The question essentially asks: if \(n\) is either 6 or 5, what is the value of \(n\)?
(1) \((n - 6)(n - 7) = 0\).
This statement implies that \(n\) is either \(6\) or \(7\). Therefore, \(n\) cannot be 5, and it must be 6. Sufficient.
(2) \((n - 5)(n - 3) ≠ 0\).
This statement implies that \(n\) is neither \(5\) nor \(3\). Since \(n\) is not 5, it must be 6. Sufficient
Answer: D
