Digital root

The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. For example, in base 10, the digital root of the number 12345 is 6 because the sum of the digits in the number is 1 + 2 + 3 + 4 + 5 = 15, then the addition process is repeated for the resulting number 15, so that the sum of 1 + 5 equals 6, which is the digital root of that number. In base 10, this is equivalent to taking the remainder upon division by 9 (except when the digital root is 9, where the remainder upon division by 9 will be 0), which allows it to be used as a divisibility rule. The formula for the function d r b : N → ⋃ k = 0 b − 1 ⁡ { k } , b ∈ N ⩾ 2 {\displaystyle \mathrm {dr} _{b}:\mathbb {N} \rightarrow \mathop {\bigcup } _{k=0}^{b-1}\{k\},\quad b\in \mathbb {N} _{\geqslant 2}} is expressed as: d r b ( n ) := { 0 n = 0 1 + ( ( n − 1 ) mod ( b − 1 ) ) n > 0 {\displaystyle \mathrm {dr} _{b}(n):={\begin{cases}0&n=0\\1+((n-1){\bmod {(}}b-1))&n>0\end{cases}}} .