Determine whether or not a binary number is divisible by $3$

Question

Let $K$ be a natural number with $n$ binary digits.

Is there an $O(n)$ method for deciding whether or not $K$ is divisible by $3$?

$3|K \iff d_1-d_2+d_3-d_4\dots\pm d_n=0$ works correctly up to $20$, but fails for $21$.

Answer 1

As $2\equiv-1\pmod3,$

$$\sum_{r=0}^n2^rb_r\equiv \sum_{r=0}^n(-1)^rb_r\pmod3$$ which should work for any integer $n>0$ where $b_r$ are the bits, hence $0\le b_r\le1\forall r$

Answer 2

Another way to prove the accepted answer without using the congruence relation. First, note that we have: $$ \frac{1}{3} \text{ = } 0+\frac{1}{3} \\ \frac{2}{3} \text{ = } 1-\frac{1}{3} \\ \frac{4}{3} \text{ = } 1+\frac{1}{3} \\ \frac{8}{3} \text{ = } 3-\frac{1}{3} \\ \\ \vdots $$ And, generally: $$ \frac{2^k}{3}= \begin{cases} p - \frac{1}{3} & \text{if } k \text{ is odd} \\ q + \frac{1}{3} & \text{if } k \text{ is even} . \end{cases} $$ Where $p$ and $q$ are some integers. It can be proved using induction. In order to check whether a binary integer $N$ with $n$ digits is divisble by $3$ we can write $\frac{N}{3}$ as follows:

$$ \begin{split} \frac{N}{3} = \frac{\sum_{i=0}^{n-1}b_{i}2^{i}}{3} = (\frac{2^0}{3}b_0+\frac{2^2}{3}b_2+\frac{2^4}{3}b_4+\ldots+\frac{2^{2r}}{3}b_{2r}) \\ + (\frac{2^1}{3}b_1+\frac{2^3}{3}b_3+\frac{2^5}{3}b_5+\ldots+\frac{2^{2s-1}}{3}b_{2s-1}). \end{split} $$ Where $r= \left \lfloor \frac{n-1}{2} \right \rfloor$, $s=\left \lfloor \frac{n}{2} \right \rfloor$. Now replacing the $\frac{2^k}{3}$ terms yields: $$ \begin{align} \frac{N}{3} & = ((q_0+\frac{1}{3})b_0+(q_2+\frac{1}{3})b_2+\ldots+(q_{2r}+\frac{1}{3})b_{2r}) \\ & \quad + ((p_1-\frac{1}{3})b_1+(p_3-\frac{1}{3})b_3+\ldots+(p_{2s-1}-\frac{1}{3})b_{2s-1}) \\ & = l + \frac{b_0-b_1+b_2-b_3+\ldots+(-1)^{n-1}b_{n-1}}{3}. \end{align} $$

Where $l$ is the sum of $p_ib_i$s and $q_ib_i$s and is already an integer. So in order for $N$ to be divisible by $3$ (i.e., the quotient to be an integer), $b_0-b_1+b_2-b_3+\ldots+(-1)^{n-1}b_{n-1}$ must be divisible by $3$.