Definition:Digital Root

Definition

Let $n \in \Z: n \ge 0$.

Let $n$ be expressed in base $b$ notation.

Let $n_1 = s_b \left({n}\right)$ be the digit sum of $n$ to base $b$.

Then let $n_2 = s_b \left({n_1}\right)$ be the digit sum of $n_1$ to base $b$.

Repeat the process, until $n_m$ has only one digit, i.e. that $1 \le n_m < b$.

Then $n_m$ is the digital root of $n$ to the base $b$.

Examples

In conventional base 10 notation, we have:

$s_{10} \left({34716}\right) = 3 + 4 + 7 + 1 + 6 = 21$

and then:

$s_{10} \left({21}\right) = 2 + 1 = 3$.

So the digital root of $34716$ (base $10$) is $3$.

In binary notation, we have:

$s_{2} \left({10010111101_2}\right) = 1 + 0 + 0 + 1 + 0 + 1 + 1 + 1 + 1 + 0 + 1 = 7 = 111_2$

$s_{2} \left({111_2}\right) = 1 + 1 + 1 = 3 = 11_2$

$s_{2} \left({11_2}\right) = 1 + 1 = 2 = 10_2$

$s_{2} \left({10_2}\right) = 1 + 0 = 1 = 1_2$

It is pretty obvious that the digital root of any number in base $2$ is always $1$.