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16 Sep 2014, 00:46
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If \(n\) and \(p\) are positive integers, is \(10^n - 1\) divisible by \(p\)?
(1) \(p\) is not divisible by either 5 or 2.
(2) \(p\) is not divisible by 9.
(1) \(p\) is not divisible by either 5 or 2.
(2) \(p\) is not divisible by 9.
16 Sep 2014, 00:46
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Official Solution:
If \(n\) and \(p\) are positive integers, is \(10^n - 1\) divisible by \(p\)?
Before examining the statements, note that since the sum of the digits of \(10^n - 1\), where \(n\) is a positive integer, is 9, it will always be divisible by 9.
(1) \(p\) is not divisible by either 5 or 2.
If \(n = 1\) and \(p = 3\), then the answer is YES. However, if \(n = 1\) and \(p = 7\), then the answer is NO. This statement is not sufficient.
(2) \(p\) is not divisible by 9.
If \(n = 1\) and \(p = 3\), then the answer is YES. However, if \(n = 1\) and \(p = 7\), then the answer is NO. This statement is not sufficient.
(1) + (2) The same examples used in both statements are still valid when combining the statements: if \(n = 1\) and \(p = 3\), then the answer is YES. However, if \(n = 1\) and \(p = 7\), then the answer is NO. Thus, combining the statements is not sufficient.
Answer: E
If \(n\) and \(p\) are positive integers, is \(10^n - 1\) divisible by \(p\)?
Before examining the statements, note that since the sum of the digits of \(10^n - 1\), where \(n\) is a positive integer, is 9, it will always be divisible by 9.
(1) \(p\) is not divisible by either 5 or 2.
If \(n = 1\) and \(p = 3\), then the answer is YES. However, if \(n = 1\) and \(p = 7\), then the answer is NO. This statement is not sufficient.
(2) \(p\) is not divisible by 9.
If \(n = 1\) and \(p = 3\), then the answer is YES. However, if \(n = 1\) and \(p = 7\), then the answer is NO. This statement is not sufficient.
(1) + (2) The same examples used in both statements are still valid when combining the statements: if \(n = 1\) and \(p = 3\), then the answer is YES. However, if \(n = 1\) and \(p = 7\), then the answer is NO. Thus, combining the statements is not sufficient.
Answer: E
23 Jul 2016, 03:10
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I think this is a high-quality question and I agree with explanation. This is a great question but must be a 700 level question and not 600 level. Not sure how you classified this as 600
