Question

Given a positive integer N, consider the following process: Let `S(N)` denote the sum of the digits of N. Take the sum of the digits of `S(N)`. Repeat this operation until a single-digit number is obtained. We call the number of times we performed the above process until we obtained a single-digit number the "depth" of N. For example, the depth of 49 is `S(49)=13 -> S(13)=4`, the operation was performed twice (and the depth of 45 is 1).

a) Prove that for every number N there is indeed a finite depth, that is, a single-digit number is always obtained at some stage of the process.

b) Let `x(n)` denote the minimum number (with the smallest value) with depth N. Find the remainder of `x(5776)` when divided by 6. Justify your answer!

c) Find the remainder of the number `x(5776) - x(5708)` when divided by 2016. Justify your answer!