Which of the following cannot be true if the sum of k consecutive integers is 0?
I. The product of the k integers is positive
II. The smallest of the k integers is zero
III. The largest of the k integers is negative
Choose the correct choice from the following:
A) I only
B) II only
C) III only
D) I and III
E) I, II and III
For the sum of k consecutive integers to be 0 we must have one of the following cases: {0}, {-1, 0, 1}, {-2, -1, 0, 1, 2}, ..., {..., -3, -2, -1, 0, 1, 2, 3, ...}.   I. The product of the k integers is positive. As you can see the product of the integers in either of these sets is always 0, thus I cannot be true.   II. The smallest of the k integers is zero. If the set is {0}, then it's possible.   III. The largest of the k integers is negative. As you can see the largest of the integers is always more than or equal to zero, thus III cannot be true.   Answer: I and III.
<result>D</result>
Which of the following cannot be true if the sum of k consecutive integers is 0?
I. The product of the k integers is positive
II. The smallest of the k integers is zero
III. The largest of the k integers is negative