Example 9 On the blackboard, there are two positive integers, one is 2002, and the other is a number less than 2002. If the average of these two numbers $m$ is an integer, then the following operation can be performed: one of the numbers is erased and replaced by $m$. How many times can such an operation be performed at most?
Without loss of generality, let these two positive integers be $a, b(a>b)$.
Consider the absolute value of the difference between the two numbers.
Initially, it is $|a-b|$. After the first operation, the absolute value of the difference between the two numbers is $\left|a-\frac{a+b}{2}\right|=\left|\frac{a-b}{2}\right|$. Proceeding in this manner, after each operation, the absolute value of the difference between the two numbers becomes half of the original absolute value of the difference, and the average $m$ of the two numbers is an integer. Therefore, the number of allowed operations $n$ should satisfy
$2^{n}<2002$.
Clearly, the maximum value of $n$ is 10, meaning the operation can be performed at most 10 times.

At this point, the initial two positive integers on the blackboard are 2002 and 978.