M05-22

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