number theory - Lucas' Theorem and Pascal's Triangle

I have a general question about Lucas' Theorem. Lucas' Theorem says the following:

Theorem (Lucas' Theorem) Let $p$ be a prime number. Write $n$ and $k$ in base $p$: $n = a_0 + a_{1}+a_{2}p^{2} + \cdots + a_{d}p^{d}$ and $k = b_0+b_{1}p+b_{2}p^{2}+ \cdots +b_{d}p^{d}$ where $0 \leq a_i,b_i \leq p-1$. Then $$ \ \binom{n}{k} \equiv \prod_{i=0}^{d} \binom{a_i}{b_i} \pmod{p}$$

Another way of viewing this is that in $\mathbb{Z}_p$, $\binom{n}{k}$ is a product of binomial coefficients.

Question. How does this theorem correspond to the geometric interpretation of Pascal's Triangle? Namely, for any entry in Pascal's triangle which is odd mark a $\text{x}$. Else leave it blank. So we basically get a fractal like structure. Is Lucas' Theorem just a statement of a way to get extra rows of Pascal's Triangle given that we know $2^n$ rows?