# digital root

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. 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 $m-(b-1)\lfloor{{m-1}\over{b-1}}\rfloor$.

Title digital root DigitalRoot 2013-03-22 15:59:34 2013-03-22 15:59:34 PrimeFan (13766) PrimeFan (13766) 13 PrimeFan (13766) Definition msc 11A63 repeated digit sum repeated digital sum additive persistence