The Rhind papyrus contained a table with representations as sums of unit fractions for each of the fractions $2/n$, where $n$ is an odd number from $3$ to $101$. For example, $$ \frac{2}{5} = \frac{1}{3} + \frac{1}{15}. $$
Why were these representations enough for the Egyptians to get a representation as a sum of unit fractions for every irreducible fraction $p/q$?
I read several online materials, but couldn't see why this is the case. Many of those documents say that the reason is that Egyptians used the binary representation to multiple numbers, but how exactly this could help? For instance, to represent $\frac{13}{17}$ with unit fractions, you can use that $13 = 8 + 4 +1$, so $\frac{13}{17} = 4\frac{2}{17} + 2.\frac{2}{17} + \frac{1}{17}$ and then you can use the table, but how can we get our final representation as a sum of unit fractions having different denominators?