Hi,
Statement 1 says that the product of any two numbers is zero.
This could be true if the numbers are all zero: 0, 0, 0.
It could also be true if one of the numbers is non-zero and the remainder are zero: 0, 0, 7.
(Every time we choose two, at least one of them will be zero. Since the product of zero and any other number is zero, we are okay with one non-zero number. We would NOT be okay with two non-zero numbers, since their product would be non-zero.)
Statement 2 says that the sum of any two numbers in the list is equal to 0.
This can clearly be the case if all of the numbers are zero: 0, 0, 0.
Let's try to build a list of more than two numbers that meets the condition and that contains a non-zero number:
Let's start with the non-zero number 2. Note that the only number we can add to 2 that gives us a sum of 0 is (-2).
As long as the list has only two numbers, we are in good shape: 2, -2. The sum of any two numbers on this list is zero. But we are told that the list has more than two numbers. What happens when we add a third number?
If we add another 2, we will have 2, 2, -2. This creates a problem, since 2 + 2 = 4.
If we add another -2, we will have 2, -2, -2. This also creates a problem, since -2 + -2 = -4.
So we can't add either 2 or -2. We also can't add any other number to the list and have each pair of numbers sum to 0.
This same reasoning will apply for any non-zero number we choose.
Algebraically, we could say that if the first non-zero number is N, the second must be -N. If we add a third number X, the list is N, -N, X.
According to the rule, N + X = 0 must be true, AND (-N) + X = 0 must be true.
So, X = -N AND X = N. This is impossible! (Unless X and N are both zero, in which case we are back to a list that has nothing but zeros.)
Since there is no way to pick three non-zero numbers in a way that allows all three pairs of two numbers to add up to zero, Statement 2 tells us that all of the numbers in the list must be zero. The correct answer is B. [spoiler](C is incorrect because Statement 1 is not necessary to reach this conclusion.)[/spoiler]
