[net.math] divisibility properties and the digital root

markp@tekmdp.UUCP (Mark Paulin) (08/26/83)

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