Q:

Recursive digit-sum

Let the recursive digit-sum(R.D.) be defined as: continue taking the sum of digits until it becomes <10.
For example, the digit-sum of 5987 = 29, the digit-sum of 29 =11
So, R.D. of 5987 is 2.
Prove that the value of R.D. recurs after each 9 numbers i.e., R.D. of any natural numbers of the form (9.a+b) where 0≤b<9 are same.

A:

R.D of a number $n$ is nothing but the value  $n\pmod9$ which repeats after every $9th$ number. Thus, R.D sum of $n$ and $9k+n$ is same.

