16. (15 points) Replacing the larger number of two different natural numbers with their difference is called one operation. Continue this process until the two numbers are the same. For example, for 20 and 26, the process is as follows:
$(20,26) \rightarrow(20,6) \rightarrow(14,6) \rightarrow(8,6) \rightarrow(2,6) \rightarrow(2,4) \rightarrow(2,2)$
(1) Perform the above operation on 45 and 80.
(2) If the final same number obtained from two four-digit numbers through the above operation is 17, find the maximum value of the sum of these two four-digit numbers.
【Solution】Solution: $(1)(45,80) \rightarrow(45,35) \rightarrow(10,35) \rightarrow(10,25) \rightarrow(10,15) \rightarrow(10$,
5) $\rightarrow(5,5)$.
(2) $9999 \div 17=588 \cdots 3$,
$9999-3=9996$,
$9996-17=9979$,
$9996+9979=19975$.
Answer: The maximum value of the sum of the two four-digit numbers is 19975.