I was wondering why for example $\dfrac{1111_b \cdot 111_b}{11_b \cdot 1_b} = 11211_b$. Is there a good explanation for this and is there a name for products like this which have constant digits value independent of base? It also seems to hold with $\dfrac{11111_b \cdot 1111_b \cdot 111_b}{111_b \cdot 11_b \cdot 1_b}$ and other numbers as well.
Here is my general question:
Prove that $$R(m,k) = \frac{ 1^{(m)}_b \cdot 1^{(m-1)}_b\cdots 1^{(m-k+1)}_b}{1^{(1)}_b \cdot 1^{(2)}_b\cdots 1^{(k)}_b} = c_b$$ where $c$ is a constant and $b>2$.
This is because multiplication in a number system becomes a special type of convolution with carry. If the digits involved are small enough compared to the base used, then carry will never happen and we can calculate expressions like yours for example as a sequence of polynomial multiplications and divisions.
For example the first one: $1111_b$ can first be divided by $11_b$ to produce $101_b$, then the multiplication will be $111_b + 100_b\cdot 111_b = 11211_b$