%%% This file is part of PlanetMath snapshot of 2009-01-12
%%% Primary Title: multiplicative digital root
%%% Primary Category Code: 11A63
%%% Filename: MultiplicativeDigitalRoot.tex
%%% Version: 1
%%% Owner: CompositeFan
%%% Author(s): Lando47
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\begin{document}

 Given an \htmladdnormallink{integer}{http://planetmath.org/encyclopedia/RationalInteger.html} $m$ consisting of $k$ \htmladdnormallink{digits}{http://planetmath.org/encyclopedia/PlaceSystems.html} $d_1, \dots, d_k$ in \htmladdnormallink{base}{http://planetmath.org/encyclopedia/PlaceSystems.html} $b$, let $$j = \prod_{i = 1}^{k} d_i,$$ then repeat this \htmladdnormallink{operation}{http://planetmath.org/encyclopedia/Operation.html} on the digits of $j$ until $j < b$. This stores in $j$ the {\em multiplicative digital root} of $m$. The \htmladdnormallink{number}{http://planetmath.org/encyclopedia/Number.html} of \htmladdnormallink{iterations}{http://planetmath.org/encyclopedia/Iteration.html} of the \htmladdnormallink{multiplication}{http://planetmath.org/encyclopedia/Multiplication.html} operation is called the {\em multiplicative persistence} of $m$.

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