Using $p$-adic representation of a number in $\mathbb{Z}^{d}$

Question

Let $L,K >1$ and $d \ge 1$ be all integers and consider: $$\Lambda_{K} = \{0,1,2,...,L^{K}-1\}^{d} \subset \mathbb{Z}^{d}$$ This is an hypercube with cardinality $|\Lambda_{K}| = L^{dK}$. My professor said that any $x \in \Lambda_{k}$ can be written as: $$x = \sum_{k=1}^{K}\alpha_{k}L^{k-1}$$ where $\alpha_{k} \in \{0,...,L-1\}^{d}$. I don't understand this representation. How can one construct such a representation and what does it mean?

Answer 1

An element of $\Lambda_K$ is a $d$-tuple $(u_1,\dots,u_d)$ of nonnegative integers $<L^K$.

The $i$th coordinate of $\alpha_k$ will be the $k$th digit (from the right) of $u_i$ when represented in number system of base $L$.
These can be determined by using division and remainders, just as in the familiar case when $L=10$.