10. The digits from 1 to 9 are to be written in the nine cells of the $3 \times 3$ grid shown, one digit in each cell.
The product of the three digits in the first row is 12 .
The product of the three digits in the second row is 112 .
The product of the three digits in the first column is 216 .
The product of the three digits in the second column is 12 .
What is the product of the digits in the shaded cells?

A 24
B 30
С 36
D 48
E 140
SOLUTION
B

We note first that 112 is divisible by 7 , but neither 216 nor 12 is divisible by 7 . Therefore, the digit 7 is in the second row but not in the first or second columns. Hence the digit 7 is written in the cell in the second row and third column.
Because $112=7 \times 16$, the product of the other two digits in the second row is 16 . Since the digits have to be different, these digits are 2 and 8 in some order. Because 8 is not a factor of 12 , the digit 8 cannot be in the second column. Therefore 8 is written in the cell in the second row and first column, and 2 is written in the cell in the second row and second column.

Therefore we know that the second row is as shown in the diagram.
Because $216=8 \times 27$, the product of the other two digits in the first column is 27 . Hence these digits are 3 and 9 . Because 9 is not a factor of 12, the digit 9 cannot be in the first row. Hence 9 is in the first column and third row, and 3 is in the first column and first row.

The product of the three digits in the second column is 12 . One of these digits is 2 . Hence, the product of the other two digits in the second column is 6 . As 2 and 3 are already in the grid, these digits are 1 and 6 .
The digit 6 cannot be in the first row, as otherwise the product of the digits in the first row would not be 12. It follows that the digit 1 is in the first row and second column, and 6 is in the third row and second column. The digit in the first row and third column is therefore 4 . The remaining digit 5 is in the third row and third column.

Therefore the digits are written as shown in the diagram. The digits in the shaded squares are 3,2 and 5 . The product of these digits is 30 .