Digital Root

Consider the process of taking a number, taking its digit sum, then adding the digits of numbers derived from it, etc., until the remaining number has only one digit. The number of additions required to obtain a single digit from a number  in a given base is called the additive persistence of , and the digit obtained is called the digital root of .

For example, the sequence obtained from the starting number 9876 in base 10 is (9876, 30, 3), so 9876 has an additive persistence of 2 and a digital root of 3. The base-10 digital roots of the first few integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, ... (OEIS A010888). The digital root  of an integer  can be computed without actually performing the iteration using the simple congruence formula

			(1)
			(2)

REFERENCES:

Sloane, N. J. A. Sequences A007612/M1114 and A010888 in "The On-Line Encyclopedia of Integer Sequences."

Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 28, 2004. https://www.mathematicaguidebooks.org/.