18. The numbers $2,3,12,14,15,20,21$ may be divided into two sets so that the product of the numbers in each set is the same.
What is this product?
A 420
B 1260
C 2520
D 6720
E 6350400
Solution
C If all the numbers $2,3,12,14,15,20,21$ are multiplied the result will be the square of the common product of the two sets with the same product. Now
$$
\begin{aligned}
2 \times 3 \times 12 \times 14 \times 15 \times & 20 \times 21 \\
& =2 \times 3 \times\left(2^{2} \times 3\right) \times(2 \times 7) \times(3 \times 5) \times\left(2^{2} \times 5\right) \times 3 \times 7 \\
& =2^{6} \times 3^{4} \times 5^{2} \times 7^{2} \\
& =\left(2^{3} \times 3^{2} \times 5 \times 7\right)^{2} .
\end{aligned}
$$

Therefore the common product is $2^{3} \times 3^{2} \times 5 \times 7=2520$.