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

Title	digital root
Canonical name	DigitalRoot
Date of creation	2013-03-22 15:59:34
Last modified on	2013-03-22 15:59:34
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	13
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A63
Synonym	repeated digit sum
Synonym	repeated digital sum
Defines	additive persistence