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16 Sep 2014, 00:25
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If \(x\) is a positive integer, is \(x\) divisible by 15?
(1) \(x\) is a multiple of 10
(2) \(x^2\) is a multiple of 12
(1) \(x\) is a multiple of 10
(2) \(x^2\) is a multiple of 12
16 Sep 2014, 00:25
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Official Solution:
If \(x\) is a positive integer, is \(x\) divisible by 15?
(1) \(x\) is a multiple of 10.
\(x\) could be 10 (not divisible by 15) or 30 (divisible by 15). Thus, this statement alone is insufficient.
(2) \(x^2\) is a multiple of 12.
Since \(x\) is a positive integer, \(x^2\) is a perfect square. The smallest perfect square that is a multiple of 12 is 36, which implies that the least value of \(x\) is 6. However, if \(x=6\), then it is not divisible by 15, while if \(x=6*15\), it is divisible by 15. Therefore, this statement alone is insufficient.
Note that from statement (2), we can deduce that \(x\) must be a multiple of 3, as otherwise, the prime factor of 3 would not appear in \(x^2\).
(1)+(2) Since \(x\) is a multiple of both 10 and 3, it must be a multiple of their least common multiple, which is 30. Thus, \(x\) is divisible by 15. Therefore, the combination of both statements is sufficient to answer the question.
Answer: C
If \(x\) is a positive integer, is \(x\) divisible by 15?
(1) \(x\) is a multiple of 10.
\(x\) could be 10 (not divisible by 15) or 30 (divisible by 15). Thus, this statement alone is insufficient.
(2) \(x^2\) is a multiple of 12.
Since \(x\) is a positive integer, \(x^2\) is a perfect square. The smallest perfect square that is a multiple of 12 is 36, which implies that the least value of \(x\) is 6. However, if \(x=6\), then it is not divisible by 15, while if \(x=6*15\), it is divisible by 15. Therefore, this statement alone is insufficient.
Note that from statement (2), we can deduce that \(x\) must be a multiple of 3, as otherwise, the prime factor of 3 would not appear in \(x^2\).
(1)+(2) Since \(x\) is a multiple of both 10 and 3, it must be a multiple of their least common multiple, which is 30. Thus, \(x\) is divisible by 15. Therefore, the combination of both statements is sufficient to answer the question.
Answer: C
General Discussion
earnit
Joined: 06 Mar 2014
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27 Sep 2014, 13:20
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Bunuel
I couldn't follow the highlighted portion actually.
I seem to find DS Questions with given statements such as below one little difficult/time consuming:
1) x multiple of 55
2) x^3 multiple of 110
Please suggest if there are any familiar types your aware of.
Would be really grateful.
