Question 193
A Sudoku grid is a $9 \times 9$ table in which each row and each column contains each of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ in some order. In addition, the nine $3 \times 3$ subgrids contain each of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once. Suppose the product of all nine numbers on one diagonal is $M$ and the product of all nine numbers on the other diagonal is $N$.
Prove that it is possible for the product $MN$ to be divisible by $2025^3$ but it is not possible for $MN$ to be divisible by $2025^4$.
Prove that it is possible for the product $MN$ to be divisible by $2025^3$ but it is not possible for $MN$ to be divisible by $2025^4$.