Example 2 Find the greatest common divisor of $48$, $60$, and $72$.
Solve: Decompose the three numbers into prime factors respectively:
$$\begin{array}{l}
48=2 \times 2 \times 2 \times 2 \times 3=2^{4} \times 3 \\
60=2 \times 2 \times 3 \times 5=2^{2} \times 3 \times 5 \\
72=2 \times 2 \times 2 \times 3 \times 3=2^{3} \times 3^{2}
\end{array}$$

By comparing the prime factors of the above three numbers, we can see that the prime factors 2, 2, 3 (or $2^{2}, 3$) are common to all three numbers, and their product is the greatest common divisor of these three numbers:
$$2 \times 2 \times 3=12 \text { or } 2^{2} \times 3=12 \text {. }$$

For the sake of clarity, let's first discuss the greatest common divisor and the least common multiple, and then discuss the Fundamental Theorem of Arithmetic. To find the greatest common divisor of several numbers, first decompose these numbers into prime factors and write them in exponential form; then, among the common prime factors, take the powers with the smallest exponents and multiply them.