Given an integer $m$ consisting of $k$ digits $d_1, \dots, d_k$ in base $b$ , let $$j = \sum_{i = 1}^{k} d_i,$$ then repeat this operation on the digits of $j$ until $j < b$ . This stores in $j$ the digital root of $m$ . The number of iterations of the sum operation is called the additive persistence of $m$ .

The digital root of $b^x$ is always 1 for any natural $x$ , while the digital root of $yb^n$ (where $y$ is another natural number) is the same as the digital root of $y$ . This should not be taken to imply that the digital root is necessarily a multiplicative function.

The digital root of an integer of the form $n(b - 1)$ is always $b - 1$ .

Another way to calculate the digital root for $m > b$ is with the formula $m - (b - 1)\lfloor {{m - 1} \over {b - 1}} \rfloor$ .