6. Each of the factors of 100 is to be placed in a 3 by 3 grid, one per cell, in such a way that the products of the three numbers in each row, column and diagonal are all equal. The positions of the numbers 1,2, 50 and $x$ are shown in the diagram.
What is the value of $x$ ?
A 4
B 5
C 10
D 20
E 25
Solution
D
Let $P$ be the common product of the three numbers in each row. Then $P \times P \times P$ is the product of all the numbers in all three rows. Therefore $P^{3}$ is the product of all the factors of 100 .
Because $100=2^{2} \times 5^{2}$, it has the nine factors $1,2^{1}=2,2^{2}=4,5^{1}=5,2^{1} 5^{1}=10,2^{2} 5^{1}=20$, $5^{2}=25,2^{1} 5^{2}=50$ and $2^{2} 5^{2}=100$. The product of these factors is
$$
\begin{aligned}
1 \times 2^{1} \times 2^{2} \times 5^{1} \times 2^{1} 5^{1} \times 2^{2} 5^{1} \times 2^{1} 5^{2} \times 5^{2} \times 2^{2} 5^{2} & =2^{1+2+1+2+1+2} \times 5^{1+1+1+2+2+2} \\
& =2^{9} \times 5^{9}(=1000000000) .
\end{aligned}
$$

It follows that
$$
P^{3}=2^{9} \times 5^{9}
$$
and therefore $P=2^{3} \times 5^{3}=1000$.
From the first row we have
$$
x \times 1 \times 50=1000
$$
and hence $x=20$.