Joãozinho wrote the numbers from 1 to 100000 on the board, then started replacing each number with the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234 we get $7+2+3+4=16$ and $1+6=7$.

(a) What number is in the place of the number 98765?

(b) How many times does the number 8 appear in the final list?

(c) Which number appears the most times?
(a) $98765 \longrightarrow 9+8+7+6+5=35 \longrightarrow 3+5=8$.

(b) Let's observe that a number and the sum of its digits leave the same remainder when divided by 9. Thus, after completing the entire process, we will obtain a list of the form

$$
1,2,3,4,5,6,7,8,9,1,2,3,4, \ldots, 8,9,1
$$

Thus, up to 99999, each digit appears 11111 times, in particular, the digit 8 appears 11111 times.

(c) From the previous item, it is clear that the number that repeats the most is 1, as it appears 11112 times in the list.