21. Seven different single-digit numbers are written in the circles of the diagram shown with one number in each circle. The product of the three numbers in each of the three lines of three numbers is the same. Which number is written in the circle containing the question mark?
A 2
B 3
C 4
D 6
E 8
Solution
A

Consider the number 5. If it were used then, wherever it was placed, there would be at least one line to which it belongs and at least one to which it doesn't. The product of those lines could not be equal. The same argument applies to 0 and to 7 .
So the seven entries must be $1,2,3,4,6,8$ and 9 and their product is $1 \times 2 \times 3 \times 4 \times 6 \times 8 \times 9=2^{7} \times 3^{4}$.
The product of the numbers in the two horizontal rows must be a perfect square (the square of the common product). So the bottom digit must either be 2 or $2^{3}=8$ to account for the difference.
But it cannot be 8 because then the product of each row would be $\sqrt{2^{4} \times 3^{4}}$, which is 36 . But this is not a multiple of 8 .

So the number in the circle is 2 .
This is possible, with the first row $1,9,8$, second row $3,4,6$ and column $9,4,2$, with common product 72 .