MathDB

Let

n>1

be a positive integer. Each cell of an

n\times n

table contains an integer. Suppose that the following conditions are satisfied:
[*] Each number in the table is congruent to

1

modulo

n

. [*] 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 number in the

j^{\text{th}}

column. Prove that the sums R_1+\hdots R_n and C_1+\hdots C_n are congruent modulo

n^4

Problem Statement