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 $$1$$.