19. Design a packaging box with a square base to transport four different sizes of chess sets, where the base of each chess box is also square, with side lengths of 21 cm, 12 cm, 14 cm, and 10.5 cm, respectively. To ensure that the packaging box can completely cover the base regardless of which size of chess set it is transporting, what is the minimum side length of the packaging box's base in centimeters?
19. Solution: To ensure that each type of chess box can completely cover the bottom of the packaging box, the side length of the bottom of the packaging box should be a common multiple of the side lengths of the bottom of each chess box. Therefore, the smallest side length of the box bottom is the least common multiple of the side lengths of the bottoms of the various chess boxes. We use millimeters as the unit of length to ensure that all side lengths are integers. Using the method of prime factorization, we get
$$\begin{array}{l}
210=2 \times 3 \times 5 \times 7 \\
120=2^{3} \times 3 \times 5 \\
140=2^{2} \times 5 \times 7 \\
105=3 \times 5 \times 7
\end{array}$$

So $\{210,120,140,105\}=2^{3} \times 3 \times 5 \times 7=840$.
Answer: The minimum side length of the box bottom is 840 millimeters, which is 84 centimeters.