Final answer:
The largest number of zeros in a 3×3 table where the sum of the numbers in any row and column is zero and the number of zeros is even is six. This allows for three pairs of numbers, one positive and one negative, to be placed in the grid to balance out the remaining cells and maintain the properties of the sum.
Step-by-step explanation:
To determine the largest number of zeros in a 3×3 table where the sum of numbers in any row and column is zero and the number of zeros is even, we can logically work through the possibilities. The key information we have is that the sum of any row or column is zero.
If we were to fill the table with eight zeros, the remaining cell would also have to be zero to satisfy the condition that each row and column sum to zero. However, this would make the number of zeros odd, which is not allowed according to the question. Hence, option D is eliminated.
With six zeros, there must be three remaining numbers that need to sum to zero in each row and column. This is possible, for instance, by placing three 1's and three -1's appropriately, or any variant with equal positive and negative values. Therefore, six zeros is a possible configuration, which makes answer B (6 zeros) the correct answer.
With four zeros, the configuration can easily be achieved with the remaining five numbers balancing out to sum zero in each row and column. Hence, four is a possible but not the largest number.
With two zeros, the task is trivial, but again this would not be the largest number of zeros. Therefore, the correct answer is B (six zeros).