I believe that the correct answer to this problem is choice B.
Let's try to break statement (1): n is divisible by 3
Suppose n odd that is 9: it is divisible by 3.
Suppose n is 12: it is also divisible by 3.
Hence, statement (1) provides us with more than one answer, so it is insufficient.
Let's try to break statement (2): 2n is divisible by twice as many positive integers as n
Suppose n is odd that is 9: 1, 3, 9. It has three divisors.
Hence, 2n i.e. 2 x 9 = 18; 1, 2, 3, 6, 9, 18. It has six divisors.
Suppose n is even that is 12: 1, 2, 3, 4, 6, 12. It has six divisors.
Hence 2n i.e. 2 x 12 = 24; 1, 2, 3, 4, 6, 8, 12, 24. It has eight divisors.
Therefore statement (2) maintains the condition for the n to be odd, it is sufficient to prove that n is an odd integer.
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GMAT Assassin