If the product of a pair of integers is to be odd, both of the integers must be odd. If either of the integers are even, the product will be even.

Statement (1) is sufficient. If one of the integers is even, the product is even, so the answer is "no."

Statement (2) is also sufficient. If the sum of the integers is odd, one of the integers is odd and one of the integers is even. Again, one of the integers is even, so the product must be even. Choice (D) is correct.