MathDB

Denote by

S(n)

the sum of digits of

n

. Given a positive integer

N

, we consider the following process: We take the sum of digits

S(N)

, then take its sum of digits

S(S(N))

, then its sum of digits

S(S(S(N)))

... We continue this until we are left with a one-digit number.
We call the number of times we had to activate

S(\cdot)

the depth of

N

.
For example, the depth of 49 is 2, since

S(49)=13\rightarrow S(13)=4

, and the depth of 45 is 1, since

S(45)=9

.
[*] Prove that every positive integer

N

has a finite depth, that is, at some point of the process we get a one-digit number. [*] Define

x(n)

to be the minimal positive integer with depth

n

. Find the residue of

x(5776)\mod 6

. [*] Find the residue of

x(5776)-x(5708)\mod 2016

Problem Statement