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From: markp@tekmdp.UUCP (Mark Paulin)
Newsgroups: net.math
Subject: divisibility properties and the digital root
Message-ID: <2166@tekmdp.UUCP>
Date: Fri, 26-Aug-83 14:15:24 EDT
Article-I.D.: tekmdp.2166
Posted: Fri Aug 26 14:15:24 1983
Date-Received: Mon, 29-Aug-83 00:36:11 EDT
Lines: 32


The sum of the digits of a number N is called the "digital root" of N, dr(N).

One way to show that, in base k, N = dr(N) mod (k-1) is to apply the so-called
"synthetic division" algorithm to [the formal polynomial in k which is defined
by the base k representation of a number, and (k-1)].  To wit:

When we divide the polynomial

N = a(n)k**n + a(n-1)k**(n-1) +...+ a(2)k**2 + a(1)k + a(0)

by (k-1) using synthetic division we obtain

N = Q(k-1) + R

where

R = a(n) + a(n-1) +...+ a(2) + a(1) + a(0) = dr(N)

thus

N = dr(N) mod (k-1)

////.

The same argument, mutatis mutandis, will show that a number N in base k is
divisible by (k+1) if and only if the alternating sum of its digits is
divisible by (k+1).


Mark Paulin
...tektronix!tekmdp!markp