= Solution
Suppose two <base-p expansions> represent the same nonnegative <integer>. Pad the shorter one by trailing zero digits so that both have the same length. Reducing modulo $p$ shows that their first digits satisfy $k_0\equiv k'_0\pmod p$. Since both digits lie between $0$ and $p-1$, they are equal as <integers>.
Subtract that common digit and divide by $p$. The remaining equality is an equality of two <base-p expansions> with one fewer digit. Repeating this argument shows that every pair of corresponding digits agrees. Equivalently, repeated <Euclidean division> recovers the digits as remainders. \b[The digits are unique up to padding by zeroes.] The finite nonnegative-digit representation presupposes $k\ge0$; a negative <integer> has no such expansion.
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