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Does the integer k have at least three different positive prime factors ?

(1) k/15 is an integer
(2) k/10 is an integer

We need to determine whether k has at least three different prime factors.

Statement One Alone:

k/15 is an integer.

Statement one alone is not sufficient to answer the question. If k = 15, then k has two different prime factors (3 and 5); however, if k = 30, then k has three different prime factors (2, 3, and 5).

Statement Two Alone:

k/10 is an integer.

Statement two alone is not sufficient to answer the question. If k = 10, then k has two different prime factors (2 and 5); however, if k = 30, then k has three different prime factors (2, 3, and 5).

Statements One and Two Together:

Using statements one and two, we see that k is a multiple of both 10 and 15, and thus it is a multiple of their least common multiple, which is 30. Since all multiples of 30 have at least three different prime factors (2, 3, 5, and possibly others), the two statements together are sufficient.

Answer: C