MathDB

Oleksiy placed positive integers in the cells of the

8\times 8

chessboard. For each pair of adjacent-by-side cells, Fedir wrote down the product of the numbers in them and added all the products. Oleksiy wrote down the sum of the numbers in each pair of adjacent-by-side cells and multiplied all the sums. It turned out that the last digits of both numbers are equal to

1

. Prove that at least one of the boys made a mistake in the calculation.
For example, for a square

3\times 3

and the arrangement of numbers shown below, Fedir would write the following numbers:

2, 6, 8, 24, 15, 35, 2, 6, 8, 20, 18, 42

, and their sum ends with a digit

6

; Oleksiy would write the following numbers:

3, 5, 6, 10, 8, 12, 3, 5, 6, 9, 9, 13

, and their product ends with a digit

0

.
\begin{tabular}{| c| c | c |} \hline 1 & 2 & 3 \\ \hline 2 & 4 & 6 \\ \hline 3 & 5 & 7 \\ \hline \end{tabular}
Proposed by Oleksiy Masalitin and Fedir Yudin

Problem Statement