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16 Sep 2014, 00:17
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If a positive integer \(x\) has a remainder of 2 when divided by 5, is \(x\) a multiple of 7?
(1) \(x\) is a prime number
(2) \(x + 3\) is a multiple of 10
(1) \(x\) is a prime number
(2) \(x + 3\) is a multiple of 10
16 Sep 2014, 00:17
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Official Solution:
If a positive integer \(x\) has a remainder of 2 when divided by 5, is \(x\) a multiple of 7?
(1) \(x\) is a prime number.
Since \(x\) is a prime number and 2 more than a multiple of 5, possible values for \(x\) are 2, 7, 17, 37, ... If \(x = 7\), it is a multiple of 7, but for other values, it is not. Insufficient.
(2) \(x + 3\) is a multiple of 10
Given that \(x\) is 2 more than a multiple of 5 and 3 less than a multiple of 10, possible values for \(x\) are 7, 17, 27, 37, 47, 57, 67, 77, ... If \(x\) is 7 or 77, it is a multiple of 7, but if it is 17 or 37, it is not. Insufficient.
(1)+(2) Considering both statements, \(x\) can still be 7, which would mean it is a multiple of 7, as well as other values, such as 17 or 37, which are not multiples of 7. Not sufficient.
Answer: E
If a positive integer \(x\) has a remainder of 2 when divided by 5, is \(x\) a multiple of 7?
(1) \(x\) is a prime number.
Since \(x\) is a prime number and 2 more than a multiple of 5, possible values for \(x\) are 2, 7, 17, 37, ... If \(x = 7\), it is a multiple of 7, but for other values, it is not. Insufficient.
(2) \(x + 3\) is a multiple of 10
Given that \(x\) is 2 more than a multiple of 5 and 3 less than a multiple of 10, possible values for \(x\) are 7, 17, 27, 37, 47, 57, 67, 77, ... If \(x\) is 7 or 77, it is a multiple of 7, but if it is 17 or 37, it is not. Insufficient.
(1)+(2) Considering both statements, \(x\) can still be 7, which would mean it is a multiple of 7, as well as other values, such as 17 or 37, which are not multiples of 7. Not sufficient.
Answer: E
21 Mar 2023, 07:48
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
