Question 27
Let $n \gt 1$ be a positive integer. Each cell of an $n \times n$ table contains an integer. Suppose that the following conditions are satisfied:
(i) Each number in the table is congruent to $1$ modulo $n$;
(ii) The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^{2}$.
Let $R_{i}$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_{j}$ be the product of the numbers in the $j^{\text{th}}$ column. Prove that the sums $R_{1} + \cdots + R_{n}$ and $C_{1} + \cdots + C_{n}$ are congruent modulo $n^{4}$.
(i) Each number in the table is congruent to $1$ modulo $n$;
(ii) The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^{2}$.
Let $R_{i}$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_{j}$ be the product of the numbers in the $j^{\text{th}}$ column. Prove that the sums $R_{1} + \cdots + R_{n}$ and $C_{1} + \cdots + C_{n}$ are congruent modulo $n^{4}$.