# Paper 4, Section II, E

Let $p$ be a prime. A base $p$ expansion of an integer $k$ is an expression

$k=k_{0}+p \cdot k_{1}+p^{2} \cdot k_{2}+\cdots+p^{\ell} \cdot k_{\ell}$

for some natural number $\ell$, with $0 \leqslant k_{i} for $i=0,1, \ldots, \ell$.

(i) Show that the sequence of coefficients $k_{0}, k_{1}, k_{2}, \ldots, k_{\ell}$ appearing in a base $p$ expansion of $k$ is unique, up to extending the sequence by zeroes.

(ii) Show that

$\left(\begin{array}{l} p \\ j \end{array}\right) \equiv 0 \quad(\bmod p), \quad 0

and hence, by considering the polynomial $(1+x)^{p}$ or otherwise, deduce that

$\left(\begin{array}{c} p^{i} \\ j \end{array}\right) \equiv 0 \quad(\bmod p), \quad 0

(iii) If $n_{0}+p \cdot n_{1}+p^{2} \cdot n_{2}+\cdots+p^{\ell} \cdot n_{\ell}$ is a base $p$ expansion of $n$, then, by considering the polynomial $(1+x)^{n}$ or otherwise, show that

$\left(\begin{array}{l} n \\ k \end{array}\right) \equiv\left(\begin{array}{l} n_{0} \\ k_{0} \end{array}\right)\left(\begin{array}{l} n_{1} \\ k_{1} \end{array}\right) \cdots\left(\begin{array}{l} n_{\ell} \\ k_{\ell} \end{array}\right) \quad(\bmod p)$