Example 1 Find the greatest common divisor of 36 and 24.
Solve: Decompose these two numbers into prime factors
$$36=2 \times 2 \times 3 \times 3, \quad 24=2 \times 2 \times 2 \times 3$$

By comparing the prime factors of these two numbers, we can see that the prime factors $2,2,3$ are common to both numbers. Their product is the greatest common divisor of these two numbers:
$$2 \times 2 \times 3=12$$

To find the greatest common divisor of several positive integers, first decompose these positive integers into prime factors, then take the common prime factors (the same prime factors are taken according to the number of times they are common) and multiply them.