Question 46
By writing the digits $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$ and $9$ in the cells of a $3 \times 3$ board, without repetitions, $6$ numbers of $3$ digits each are formed: one in each row and one in each column. For instance, if the board is filled in like in this picture
| | Column 1 | Column 2 | Column 3 |
|---------|----------|----------|----------|
| Row 1 | 1 | 2 | 7 |
| Row 2 | 5 | 6 | 3 |
| Row 3 | 4 | 9 | 8 |
then the $6$ numbers are: $127$, $563$, $498$, $154$, $269$ and $738$.
We have to fill in the $3 \times 3$ board so that the number in the first row is a multiple of $2$, the number in the second row is a multiple of $3$, the number in the third row is a multiple of $4$, the number in the first column is a multiple of $5$, the number in the second column is a multiple of $6$, and the number in the third column is a multiple of $7$.
Determine all possible ways to fill in the board.
| | Column 1 | Column 2 | Column 3 |
|---------|----------|----------|----------|
| Row 1 | 1 | 2 | 7 |
| Row 2 | 5 | 6 | 3 |
| Row 3 | 4 | 9 | 8 |
then the $6$ numbers are: $127$, $563$, $498$, $154$, $269$ and $738$.
We have to fill in the $3 \times 3$ board so that the number in the first row is a multiple of $2$, the number in the second row is a multiple of $3$, the number in the third row is a multiple of $4$, the number in the first column is a multiple of $5$, the number in the second column is a multiple of $6$, and the number in the third column is a multiple of $7$.
Determine all possible ways to fill in the board.