15.2 Digital root 425
15.2 Digital root ..................
Given a positive integern, letd(n)be the sum of the digits ofn. If the
operationdis repeated indefinitely, it stabilizes after a finite number of
steps and yield a number between 1 and 9, which we call the digital root
ofn, denotedD(n). See [Dudeny,Amusements, p.157].
Theorem 15.1. 1.D(m+n)=D(D(m)+D(n)).
2.D(mn)=D(D(m)D(n)).3.D(mn)=D(D(m)n).4.D(D(n)) =D(n).5.D(n+9)=D(n).6.D(9n)=9.Proof.(5)D(n+9) =D(D(n)+D(9)) =D(D(n)+9) =D(n)since
D(n)is a single-digit number.
(6)D(9n)=D(9D(n)) = 9sinceD(n)has one single digit.