We want to know if the product of a certain set of integers is positive.

Statement 1 says that the product of the greatest and smallest numbers in the set is positive. That means that the greatest and smallest numbers are either both negative or both positive, and so the numbers in the set are either all positive or all negative.
So, if your set of numbers is {1, 2, 3, 5}, then the product is positive.
But if your set is {-3, -2, -1} then the product is negative.
Statement one is insufficient.

Statement 2 says that there is an even number of numbers in the set.
If your set is {2, 4, 5, 6} then the product is positive.
If your set is {-3, 4, 5, 6} then the product is negative.
Statement two is insufficient.

Put the statements together, and we see that two things must be true:
a) the highest and lowest number are BOTH positive or BOTH negative
AND
b) there is an even number of numbers in the set.

If the highest and lowest numbers are both positive, then all of the numbers in the set are positive, so the product will be positive.

If the highest and lowest are both negative, then all the numbers in the set are negative, so we just need to have an even number of them for the negatives in the product to "cancel out" and become positive.
For example: {-6, -5, -3, -1}

Any four negative numbers, or any even number of negative numbers, will always multiply to a positive product.

The answer is C.

Let me try:

IMO : E

Lets Rephrase the question : How many positive and how many negative numbers are there? ie., even/odd number of + and - integers.

St 1 says : greatest and smallest numbres are either positve or both negative but we dont know about the signs of othere numbers. So INSUFFICIENT.

St 2 says : list has even number of integers eg : list has only 2 or 4 or 6 or 8...integers. but here also we dont know signs of the integers. SO INSUFFICIENT.

If we combine st 1 & st 2 : greatest and smalles = positive/negative & list has even number of integers. lets say list has 6 numbers.

{1,2,3,4,5,6} = positive.
{1,2,3,4,-5,6} = negative.

so my take is E