IMO E

we need to whether number of positive integers and number of negative integers are even or odd

got my mistake. it must be C

Statement 1: The product of the greatest and smallest of the integers in the list is positive.

This means either:
- all the integers are positive, or
-all the integers are negative

if all the integers are negative, the product of all the integers in the list will be positive if there is an even number of integers, OR the product will be negative if there is an odd number of integers in the list. Insufficient.

Statement 2: There is an even number of integers in the list.

-if there are 6 integers and all of them are positive, then the product is positive.
-if there are 6 integers, with 3 of them positive and the other 3 negative, then the product is negative.

Insufficient.

Both statements together:

-if all integers are positive, then the product must be positive.
-if all integers are negative, and since we know (from statement 2) that there is an even number of integers in the list, the product will be positive.

Sufficient.

Choose C

-BM-