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Let $x$ be a base $k$ number with digits: $d_1 d_2 \cdots d_n$

(This implies the value of $x$ is $ d_1k^{n-1} +d_2k^{n-2} + \dots +d_{n-1}k+ d_n$)

We define $R(x)$ to be the "digit sum root":

If $x < k$, $R(x) = x$

Otherwise $R(x) = R(d_1 + d_2 + \dots + d_n)$

That is, we keep adding up the digits until we have a single digit. (all in base k)

What interesting properties does the function $R(x)$ have? For example, what can we say about $R(x + y)$? or $R(xy)$? and so on? How can we calculate $R$ efficiently for large $x$?

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It has all the nice properties one might desire, and that we are perhaps familiar with in the case $k=10$. Hint: $x\equiv R(x)\pmod{k-1}$. – André Nicolas Aug 14 '12 at 19:10
Look on Wikipedia under "digital root": – mjqxxxx Aug 14 '12 at 19:12
I suspect you will have something like $\varphi(x)=\displaystyle\max_{n\leq x}R(n)=O(\ln x)$ – S4M Aug 14 '12 at 19:18
@AndréNicolas - looking at base 10, one has zero divisors $(3 \times 3 = 9)$. If one were to choose base 8 or base 12 this would not be the case. – Mark Bennet Aug 14 '12 at 19:23

Hint $\rm\ R(x)\equiv x\pmod{k\!-\!1}.\:$ It's the radix $\rm\,k\,$ analog of casting out $9$'s for radix $10$ (decimal).

Remark $\ $ You can find much further discussion in prior posts on casting nines.

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