Question 22
Let $n \ge 1$ be an integer. In an $n \times n$ table, each cell is filled with an integer. Suppose the following two conditions hold:
(i) Every integer in the table leaves remainder $1$ when divided by $n$.
(ii) The sum of every row, and the sum of every column, leaves remainder $n$ when divided by $n^2$.
Let $R_i$ be the product of all numbers in the $i$-th row, and $C_j$ the product of all numbers in the $j$-th column.
Prove that $n^4$ divides $\sum_{i=1}^n R_i - \sum_{j=1}^n C_j$.
(i) Every integer in the table leaves remainder $1$ when divided by $n$.
(ii) The sum of every row, and the sum of every column, leaves remainder $n$ when divided by $n^2$.
Let $R_i$ be the product of all numbers in the $i$-th row, and $C_j$ the product of all numbers in the $j$-th column.
Prove that $n^4$ divides $\sum_{i=1}^n R_i - \sum_{j=1}^n C_j$.