I'm going to start with an example for how to derive divisibility rules for base 10 first because I feel it's easier to understand. Suppose we have $n$ which is divisible by $7$ and we want to prove it, write it in the form $n=10k+j$ so $j$. Notice that $j$ is the last digit in the decimal expansion. Now here's the trick, because $7\vert n$ we know $7\vert (n-21j)$ so $7\vert 10(k-2j)$. $10$ and $7$ are relatively prime so we get that $7\vert (k-2j)$. Put in words we can take the last digit from $n$ and subtract twice the quantity from the rest of the digits of $n$ then our new number is divisible by 7 if and only if $n$ was. For example let's check $252$:
$$252 \to 25 - 2 \times 2 = 21 \to 2 - 2 \times 1 = 0.$$
Note that you don't have to reach $0$, we could have stopped at $21$ if we recognized it's divisible by $7$.
Now we can use the same trick to come up with divisibility rules for base 2. Write $n$ in the form $$n=2k+j$$ and assume $3\vert n$. Now we have $n-3j=2(k-j)$ which is also divisible by 3, so $3\vert (k-j)$. This rule is really nice because it ends up being reduced to the alternating sum. So if the alternating sum of the binary expansion of $n$ is divisible by 3 then $n$ is. For example let's check $228;11100100$
$$
\matrix{+&-&+&-&+&-&+&-\\1&1&1&0&0&1&0&0\\ \hline \\1&-1&+1 & & & -1 & & & =0}
$$
So we see $228$ is divisible by $3$.
For divisibility by $11$ we write $n+11j=2(k+6j)$ so $11\vert(k+6j)$ if $11\vert n$. The most efficient way I can come up with to use this is to take the binary expansion of $n$, cut off all trailing $0$s and the last $1$ and add $110$ ($6$). For example $2992;101110110000$, I put vertical bars to show where I cut the number off:
$$1011101\vert10000\to 1011101+110=110001\vert 1\to 110001+110=11011\vert 1 \to$$
$$11011+110=10000\vert 1\to 10000+110=101\vert 10\to 101+110=1011$$
We reached the binary expansion for $11$ so we have shown $2992$ to be divisible by 11.
Other rules can be derived using the same trick.
