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Sum of all numbers is divisible by 13

Source: Saint Petersburg olympiad 2024, 10.1

September 22, 2024
combinatoricsnumber theory

Problem Statement

In the cells of the 2024×20242024\times 2024 board, integers are arranged so that in any 2×20232 \times 2023 rectangle (vertical or horizontal) with one cut corner cell that does not go beyond the board, the sum of the numbers is divided by 1313. Prove that the sum of all the numbers on the board is divisible by 1313.