Explanation

In this question, we can solve this question with odd and even rules, using cases to remind ourselves of the rules or clarify confusing points. Glancing at the answer choices, part of the issue is what happens when k is even or odd. If k is even, the whole thing will be even, since it only takes one even number. Therefore, (B) is out. If k is odd, then k+1 will be even, and the whole thing will be even. That fact knocks out answer choices (A) and (C). (D) doesn't seem right: what if k=6? Then k=even and the whole thing is divisible by 3, since 6 is a factor. That leaves only (E). Indeed, if k is odd, then the other two factors are even, so the product will have at least two 2's.

The correct answer is (E).