digital root

Given an integer $m$ consisting of $k$ digits $d_1,\mathrm{\dots },d_k$ in base $b$, let

$$j=\sum _{i=1}^k{d}_i,$$

then repeat this operation on the digits of $j$ until . 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 $y{b}^n$ (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 $m-(b-1)\lfloor \frac{m-1}{b-1}\rfloor$.