To solve, we need to know how many negative integers we have. If we had an odd amount, then we'll have a negative product. If it's even, then we'll have a positive product.

Statement 1: This tells us that either all of the integers are positive or all of the integers are negative. If the product of the largest and smallest is positive, that means the largest and smallest are both negative (and therefore all numbers in between) or largest and smallest are both positive (and all numbers in between).

All numbers can be negative in this instance, and we're still not sure if we have an even number of negatives or odd. Hence, this is insufficient

Statement 2: Even if the total number of integers is even, we're still not sure if we have an odd number of negative numbers. We could have a set like {-1, 1,2,3,4,5} which would be negative despite have 6 integers.

Combined: Statement 1 tells us all integers are either negative or positive. We also know that there are an even number of those integers. An even number of all negative integers would lead to a positive, so we know our answer.

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